From 80c52ee4b768cc0d1843974a4cda72f7cc6647a0 Mon Sep 17 00:00:00 2001 From: Joachim Schoeberl Date: Sat, 24 Feb 2024 10:53:34 +0100 Subject: [PATCH] Update documentation --- .buildinfo | 2 +- DG/Nitsche.html | 128 +- DG/elliptic.html | 79 +- DG/elliptic_stdDG.html | 53 +- DG/fourthorder.html | 87 +- DG/instationary.html | 71 +- DG/splitting.html | 83 +- DG/stationary.html | 77 +- DG/stokes.html | 71 +- MPIparallel/PETSc_interface.html | 38 +- MPIparallel/distmesh.html | 38 +- MPIparallel/hdivnstsimple/NavierStokesMT.html | 38 +- MPIparallel/hdivnstsimple/TestConvert.html | 38 +- .../hdivnstsimple/veclaplace-auxiliary.html | 38 +- MPIparallel/hdivnstsimple/veclaplace.html | 38 +- MPIparallel/intro.html | 38 +- MPIparallel/paralleliteration.html | 38 +- MPIparallel/parallelstokes.html | 38 +- MPIparallel/petsc.html | 38 +- MPIparallel/vectypes.html | 38 +- README.html | 38 +- ...4a0ba199b71522987f9159c6cf34c911c83779.png | Bin 0 -> 13347 bytes ...ec4699cb1fa16ec550ead237688fc31bccecda.png | Bin 0 -> 18016 bytes _sources/DG/fourthorder.ipynb | 16 +- _sources/iFEM.ipynb | 2 +- _sources/intro.md | 5 +- domaindecomposition/BDDC.html | 38 +- domaindecomposition/FETI-DP.html | 38 +- domaindecomposition/FETI.html | 38 +- domaindecomposition/introduction.html | 38 +- domaindecomposition/traces.html | 38 +- ex20210609.html | 38 +- genindex.html | 34 +- helmholtz/absorbing.html | 38 +- helmholtz/grating.html | 38 +- iFEM.html | 38 +- intro.html | 88 +- iterative/Chebyshev.html | 182 +- iterative/Richardson.html | 1502 ++++++++++++- iterative/conjugategradients.html | 123 +- iterative/gradientmethod.html | 786 ++++++- iterative/preconditioning.html | 1955 ++++++++++++++++- iterative/simple.html | 64 +- mixed/abstract.html | 88 +- mixed/abstractfem.html | 82 +- mixed/boundary.html | 100 +- mixed/pardep.html | 76 +- mixed/secondorder.html | 82 +- mixed/stokes.html | 94 +- mixedelasticity/dynamics.html | 38 +- mixedelasticity/hellingerreissner.html | 38 +- mixedelasticity/reducedsymmetry.html | 38 +- mixedelasticity/tdnns.html | 38 +- multigrid/Untitled.html | 38 +- multigrid/Untitled1.html | 38 +- multigrid/algorithms.html | 38 +- multigrid/analysisMG.html | 38 +- multigrid/analysisML.html | 38 +- multigrid/multilevel-extension.html | 38 +- multigrid/py-multigrid.html | 38 +- plates/hhj-tdnns.html | 38 +- plates/hhj.html | 38 +- plates/reissnermindlin.html | 38 +- plates/shells.html | 38 +- plates/tdnnsplate.html | 38 +- preamble.html | 38 +- primal/Untitled.html | 42 +- primal/VO Okt 6.html | 38 +- primal/boundary_conditions.html | 44 +- primal/first_example.html | 52 +- primal/solvers.html | 56 +- primal/subdomains.html | 52 +- reports/DG/splitting.err.log | 44 + saddlepoint/augmented.html | 38 +- saddlepoint/bramblepasciak.html | 38 +- saddlepoint/bramblepasciak_save.html | 38 +- saddlepoint/fewconstraints.html | 38 +- saddlepoint/parameterdependent.html | 38 +- saddlepoint/structure.html | 38 +- saddlepoint/useBP.html | 38 +- search.html | 34 +- searchindex.js | 2 +- secondorder/erroranalysis.html | 70 +- secondorder/erroranalysisl2h1.html | 60 +- secondorder/finiteelements.html | 81 +- secondorder/framework.html | 72 +- secondorder/hdiv.html | 60 +- secondorder/hybridization.html | 115 +- sobolev/Friedrichs.html | 38 +- sobolev/Untitled.html | 38 +- sobolev/Untitled1.html | 38 +- sobolev/Untitled2.html | 38 +- sobolev/poincare.html | 38 +- sobolev/trace.html | 38 +- subspacecorrection/Untitled.html | 38 +- subspacecorrection/asm.html | 38 +- subspacecorrection/examples.html | 38 +- subspacecorrection/exercise20210324.html | 38 +- subspacecorrection/minimaldd.html | 38 +- subspacecorrection/overlapping.html | 38 +- timedependent/ODEs/runge_kutta.html | 38 +- timedependent/ODEs/singlestep.html | 38 +- timedependent/intro/Untitled.html | 38 +- timedependent/intro/heatequation.html | 38 +- timedependent/intro/waveequation.html | 38 +- timedependent/parabolic/exponential.html | 38 +- timedependent/waves/Maxwell-DG.html | 38 +- timedependent/waves/Untitled.html | 38 +- timedependent/waves/lts.html | 38 +- timedependent/waves/ringresonator.html | 38 +- .../waves/testpml/ringresonator.html | 42 +- timedependent/waves/wave-leapfrogDG.html | 38 +- 112 files changed, 8827 insertions(+), 551 deletions(-) create mode 100644 _images/5a08609ce8403de46415f8b64e4a0ba199b71522987f9159c6cf34c911c83779.png create mode 100644 _images/b8f0ec331e882c9d278573cf92ec4699cb1fa16ec550ead237688fc31bccecda.png create mode 100644 reports/DG/splitting.err.log diff --git a/.buildinfo b/.buildinfo index 1360e6ff..d2108214 100644 --- a/.buildinfo +++ b/.buildinfo @@ -1,4 +1,4 @@ # Sphinx build info version 1 # This file hashes the configuration used when building these files. When it is not found, a full rebuild will be done. -config: c4a29e95c38517b30d390d674a1c5a7b +config: 22e28e0fc8da58635a9884f20d503d4d tags: 645f666f9bcd5a90fca523b33c5a78b7 diff --git a/DG/Nitsche.html b/DG/Nitsche.html index 6640fd0b..e554c0e2 100644 --- a/DG/Nitsche.html +++ b/DG/Nitsche.html @@ -8,7 +8,7 @@ - Nitsche’s Method for boundary and interface conditions — Introduction to Scientific Computing + 13. Nitsche’s Method for boundary and interface conditions — Introduction to Scientific Computing @@ -32,7 +32,7 @@ - + @@ -59,9 +59,14 @@ + + + + + @@ -123,8 +128,6 @@ - -
@@ -174,7 +177,7 @@ @@ -184,6 +187,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -332,12 +367,12 @@

    Contents

    @@ -351,7 +386,7 @@

    Contents

    -

    Nitsche’s Method for boundary and interface conditions#

    +

    13. Nitsche’s Method for boundary and interface conditions#

    from netgen.occ import *
@@ -386,6 +421,11 @@ 
    Nitsche’s Method for boundary and interface conditions
+
    error bc: 0.052752210661407825
+
    
+
    
+

    Exercises:
    -

    Nitsche’s method:#

    +

    13.1. Nitsche’s method:#

    \[ \int_\Omega \nabla u \nabla v @@ -431,6 +471,11 @@

    Nitsche’s method: +
    error bc: 3.4214848016971966e-06
+
    +
    +

    The bilinear-form is coercive w.r.t. the norm induced by \(\int_\Omega \nabla u \nabla v dx + \int_{\partial \Omega} \frac{p^2}{h} u v ds \) if \(\alpha\) is sufficiently large.

    We check positive definite by computing the few smallest eigenvalues of

    @@ -450,6 +495,15 @@

    Nitsche’s method: +
     0.985783
+ 0.987115
+ 0.990019
+ 0.991396
+ 0.995728
+
    +
    +

    Exercise:

    -

    Hybrid Interfaces#

    +

    14. Hybrid Interfaces#

    In a hybrid interface method one introduces another field \(\hat u\) only at the interface. The functions \(u\) from both sides are glued to this common interface field by a Nitsche method:

    \[ @@ -596,6 +661,13 @@

    Hybrid Interfaces +
    ('outer', 'inner', 'bar')
+('outer', 'outer', 'outer', 'outer', 'gammao', 'gammai', 'default', 'default', 'default', 'default')
+0: 0000110000
+
    +
    +

    @@ -644,6 +716,8 @@

    Hybrid Interfaces +

    @@ -677,6 +751,24 @@

    Hybrid Interfaces
    + + +
    +

    previous

    +

    12. Instationary Transport Equation

    +
    +     + +
    +

    next

    +

    15. Hybrid DG for elliptic equations

    +
    + +
    @@ -693,12 +785,12 @@

    Hybrid Interfaces +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -335,7 +370,7 @@

    Hybrid DG for elliptic equations

    -

    Hybrid DG for elliptic equations#

    +

    15. Hybrid DG for elliptic equations#

    \(\DeclareMathOperator{\opdiv}{div}\)

    The discretization of elliptic operators by DG is more tricky. Instead of the DG method, we go directly to the hybrid DG method (HDG). A notebook on standard-DG is here.

    @@ -420,6 +455,13 @@

    Hybrid DG for elliptic equations +
    ndof:  3792
+non-zero(A): 107040
+non-zero(Inv): 76416
+
    +
    +
    @@ -432,6 +474,8 @@

    Hybrid DG for elliptic equations +

    @@ -442,6 +486,9 @@

    Hybrid DG for elliptic equations +../_images/b8f0ec331e882c9d278573cf92ec4699cb1fa16ec550ead237688fc31bccecda.png +

    @@ -475,6 +522,24 @@

    Hybrid DG for elliptic equations
    + + +
    +

    previous

    +

    13. Nitsche’s Method for boundary and interface conditions

    +
    +     + +
    +

    next

    +

    16. Splitting Methods for the time-dependent convection diffusion equation

    +
    + +
    @@ -503,7 +568,7 @@

    Hybrid DG for elliptic equations - © Copyright 2023. + © Copyright 2024.

    diff --git a/DG/elliptic_stdDG.html b/DG/elliptic_stdDG.html index 213fd9d2..2265f642 100644 --- a/DG/elliptic_stdDG.html +++ b/DG/elliptic_stdDG.html @@ -32,7 +32,7 @@ - + @@ -57,6 +57,9 @@ + + + @@ -172,7 +175,7 @@     @@ -182,6 +185,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -386,6 +421,13 @@

    DG - Methods for elliptic problems +
    ndof:  2320
+non-zero(A): 88800
+non-zero(Inv): 123040
+
    +
    +
    @@ -401,6 +443,8 @@

    DG - Methods for elliptic problems +

    @@ -411,6 +455,9 @@

    DG - Methods for elliptic problems +../_images/5a08609ce8403de46415f8b64e4a0ba199b71522987f9159c6cf34c911c83779.png +

    @@ -472,7 +519,7 @@

    DG - Methods for elliptic problems - © Copyright 2023. + © Copyright 2024.

    diff --git a/DG/fourthorder.html b/DG/fourthorder.html index 4ced9023..21354745 100644 --- a/DG/fourthorder.html +++ b/DG/fourthorder.html @@ -8,7 +8,7 @@ - Fourth Order Equation — Introduction to Scientific Computing + 17. Fourth Order Equation — Introduction to Scientific Computing @@ -32,7 +32,7 @@ - + @@ -59,9 +59,14 @@ + + + + + @@ -123,8 +128,6 @@ - -
    @@ -174,7 +177,7 @@ @@ -184,6 +187,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -332,7 +367,7 @@

    Contents

    @@ -345,14 +380,14 @@

    Contents

    -

    Fourth Order Equation#

    +

    17. Fourth Order Equation#

    We consider the Kirchhoff plate equation: Find \(w \in H^2\), such that $\( \int \nabla^2 w : \nabla^2 v = \int f v \)$

    A conforming method requires \(C^1\) continuous finite elements. But there is no good option available, and thus there is no \(H^2\) conforming finite element space in NGSolve.

    -

    Hybridized \(C^0\)-continuous interior penalty method:#

    +

    17.1. Hybridized \(C^0\)-continuous interior penalty method:#

    A simple way out is to use continuous elements, and treat the missing \(C^1\)-continuity by a Discontinuous Galerkin method. A DG formulation is

    \[ @@ -371,8 +406,8 @@

    Hybridized \(C^0\)-continu

    The facet variable is the normal derivative \(n_E \cdot \nabla w\), what is oriented along the arbitrarily chosen edge normal-vector. We cannot use the FacetSpace since it does not have the orientation, but we can use the normal traces of an HDiv space. We don’t need inner basis functions, so we set order inner to 0:

    -
    import netgen.gui
-from ngsolve import *
+
    from ngsolve import *
+from ngsolve.webgui import Draw
 from netgen.geom2d import unit_square
 mesh = Mesh (unit_square.GenerateMesh(maxh=0.1))
 
    
@@ -420,6 +455,11 @@ 
    Hybridized \(C^0\)-continu
    +
    +
    <ngsolve.comp.LinearForm at 0x114c5fdb0>
+
    +
    +
    @@ -432,6 +472,11 @@

    Hybridized \(C^0\)-continu

    +
    +
    BaseWebGuiScene
+
    +
    +
    @@ -466,6 +511,24 @@

    Hybridized \(C^0\)-continu @@ -482,7 +545,7 @@

    Hybridized \(C^0\)-continu @@ -507,7 +570,7 @@

    Hybridized \(C^0\)-continu

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/DG/instationary.html b/DG/instationary.html index e069dd12..6741a92e 100644 --- a/DG/instationary.html +++ b/DG/instationary.html @@ -8,7 +8,7 @@ - Instationary Transport Equation — Introduction to Scientific Computing + 12. Instationary Transport Equation — Introduction to Scientific Computing @@ -32,7 +32,7 @@ - + @@ -59,9 +59,14 @@ + + + + + @@ -123,8 +128,6 @@ - -
    @@ -174,7 +177,7 @@ @@ -184,6 +187,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -335,7 +370,7 @@

    Instationary Transport Equation

    -

    Instationary Transport Equation#

    +

    12. Instationary Transport Equation#

    \(\DeclareMathOperator{\opdiv}{div}\)

    The time dependent transport equation is to find an \(u = u(x,t)\) such that

    @@ -372,6 +407,8 @@

    Instationary Transport Equation +

    @@ -398,6 +435,8 @@

    Instationary Transport Equation +

    @@ -456,6 +495,24 @@

    Instationary Transport Equation
    + + +
    +

    previous

    +

    11. Stationary Transport Equation

    +
    +     + +
    +

    next

    +

    13. Nitsche’s Method for boundary and interface conditions

    +
    + +
    @@ -484,7 +541,7 @@

    Instationary Transport Equation - © Copyright 2023. + © Copyright 2024.

    diff --git a/DG/splitting.html b/DG/splitting.html index 936daa7e..17934904 100644 --- a/DG/splitting.html +++ b/DG/splitting.html @@ -8,7 +8,7 @@ - Splitting Methods for the time-dependent convection diffusion equation — Introduction to Scientific Computing + 16. Splitting Methods for the time-dependent convection diffusion equation — Introduction to Scientific Computing @@ -32,7 +32,7 @@ - + @@ -62,6 +62,8 @@ + + @@ -123,8 +125,6 @@ - -
    @@ -174,7 +174,7 @@ @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -335,7 +367,7 @@

    Splitting Methods for the time-dependent convection diffusion equation

    -

    Splitting Methods for the time-dependent convection diffusion equation#

    +

    16. Splitting Methods for the time-dependent convection diffusion equation#

    \(\DeclareMathOperator{\opdiv}{div}\)

    We now want to solve a time-dependent convection diffusion equation:

    @@ -426,6 +458,11 @@

    Splitting Methods for the time-dependent convection diffusion equation

    +
    +
    <ngsolve.comp.LinearForm at 0x1169ba630>
+
    +
    +
    @@ -453,6 +490,20 @@

    Splitting Methods for the time-dependent convection diffusion equation

    +
    +
    ---------------------------------------------------------------------------
+KeyboardInterrupt                         Traceback (most recent call last)
+Cell In[4], line 15
+     13 while t < tend:
+     14     t += tau
+---> 15     aconv.Apply(gfu.vec, convu)
+     16     r.data = f.vec - convu - adiff.mat * gfu.vec
+     17     w.data = inv * r
+
+KeyboardInterrupt: 
+
    +
    +

    For higher order IMplicit-EXplicit (IMEX) methods see:

    Implicit-explicit Runge-Kutta methods for time-dependent pdes by Uri M. Ascher, Steven J. Ruuth, and Raymond J. Spiteri, Applied Numerical Mathematics, 25, 1997

    @@ -488,6 +539,24 @@

    Splitting Methods for the time-dependent convection diffusion equation
    + + +
    +

    previous

    +

    15. Hybrid DG for elliptic equations

    +
    +     + +
    +

    next

    +

    17. Fourth Order Equation

    +
    + +
    @@ -516,7 +585,7 @@

    Splitting Methods for the time-dependent convection diffusion equation - © Copyright 2023. + © Copyright 2024.

    diff --git a/DG/stationary.html b/DG/stationary.html index a9fe96ce..933d3bbd 100644 --- a/DG/stationary.html +++ b/DG/stationary.html @@ -8,7 +8,7 @@ - Stationary Transport Equation — Introduction to Scientific Computing + 11. Stationary Transport Equation — Introduction to Scientific Computing @@ -32,7 +32,7 @@ - + @@ -59,9 +59,14 @@ + + + + + @@ -123,8 +128,6 @@ - -
    @@ -174,7 +177,7 @@ @@ -184,6 +187,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -332,7 +367,7 @@

    Contents

    @@ -345,7 +380,7 @@

    Contents

    -

    Stationary Transport Equation#

    +

    11. Stationary Transport Equation#

    \(\DeclareMathOperator{\opdiv}{div}\) We consider the stationary transport equation

    @@ -364,7 +399,7 @@

    Stationary Transport Equation -

    Discontinuous Galerkin method#

    +

    11.1. Discontinuous Galerkin method#

    Finite volume methods are very popular for transport equations. Their strength is the upwind technique for a stable discretization. A DG method combines the advantages of finite elements and finite volumes. It can be seen as the extension of finite volume methods to higher order

    A DG method uses discontinuous trial and test finite element spaces, for example piecewise polynomials. It is derived as follows:

    Multiply the equation by element-wise smooth test functions, integrate:

    @@ -410,7 +445,7 @@

    Discontinuous Galerkin method -

    +
    @@ -445,7 +480,7 @@

    Discontinuous Galerkin method -

    +
    @@ -480,6 +515,24 @@

    Discontinuous Galerkin method
    + + +
    +

    previous

    +

    10. Parameter Dependent Problems

    +
    +     + +
    +

    next

    +

    12. Instationary Transport Equation

    +
    + +
    @@ -496,7 +549,7 @@

    Discontinuous Galerkin method @@ -521,7 +574,7 @@

    Discontinuous Galerkin method - © Copyright 2023. + © Copyright 2024.

    diff --git a/DG/stokes.html b/DG/stokes.html index d2b03c0c..febc94bb 100644 --- a/DG/stokes.html +++ b/DG/stokes.html @@ -8,7 +8,7 @@ - H(div)-conforming Stokes — Introduction to Scientific Computing + 18. H(div)-conforming Stokes — Introduction to Scientific Computing @@ -32,7 +32,7 @@ - + @@ -59,9 +59,14 @@ + + + + + @@ -123,8 +128,6 @@ - -
    @@ -174,7 +177,7 @@ @@ -184,6 +187,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -335,7 +370,7 @@

    H(div)-conforming Stokes

    -

    H(div)-conforming Stokes#

    +

    18. H(div)-conforming Stokes#

    \(H(div)\)-conforming hybrid DG [Lehrenfeld+Schöberl, 2016]

    Find \(u \in BDM^k, \hat u \in P^k_{t}, p \in P^k\) …

    @@ -358,7 +393,7 @@

    H(div)-conforming Stokes -
    <ngsolve.comp.Mesh at 0x1117b5210>
+
    <ngsolve.comp.Mesh at 0x114a33a10>
    @@ -424,7 +459,7 @@

    H(div)-conforming Stokes -
    BaseWebGuiScene
+
    BaseWebGuiScene
    @@ -461,6 +496,24 @@

    H(div)-conforming Stokes
    + + +
    +

    previous

    +

    17. Fourth Order Equation

    +
    +     + +
    +

    next

    +

    Application of the abstract theory

    +
    + +
    @@ -489,7 +542,7 @@

    H(div)-conforming Stokes - © Copyright 2023. + © Copyright 2024.

    diff --git a/MPIparallel/PETSc_interface.html b/MPIparallel/PETSc_interface.html index 9ba51821..54f9a8ff 100644 --- a/MPIparallel/PETSc_interface.html +++ b/MPIparallel/PETSc_interface.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@     @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +

    @@ -564,7 +596,7 @@

    PETSc preconditioner for NGSolve - © Copyright 2023. + © Copyright 2024.

    diff --git a/MPIparallel/distmesh.html b/MPIparallel/distmesh.html index c21b37ec..813e7a33 100644 --- a/MPIparallel/distmesh.html +++ b/MPIparallel/distmesh.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@     @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +

    @@ -690,7 +722,7 @@

    The ParallelDof

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/MPIparallel/hdivnstsimple/NavierStokesMT.html b/MPIparallel/hdivnstsimple/NavierStokesMT.html index 5f117ec1..a975854a 100644 --- a/MPIparallel/hdivnstsimple/NavierStokesMT.html +++ b/MPIparallel/hdivnstsimple/NavierStokesMT.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@ @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +

    @@ -491,7 +523,7 @@

    Contents

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/MPIparallel/hdivnstsimple/TestConvert.html b/MPIparallel/hdivnstsimple/TestConvert.html index aafb0450..f05c2cfc 100644 --- a/MPIparallel/hdivnstsimple/TestConvert.html +++ b/MPIparallel/hdivnstsimple/TestConvert.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@ @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -484,7 +516,7 @@

    Contents

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/MPIparallel/hdivnstsimple/veclaplace-auxiliary.html b/MPIparallel/hdivnstsimple/veclaplace-auxiliary.html index 7661f795..8e6c5a19 100644 --- a/MPIparallel/hdivnstsimple/veclaplace-auxiliary.html +++ b/MPIparallel/hdivnstsimple/veclaplace-auxiliary.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@ @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -569,7 +601,7 @@

    Contents

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/MPIparallel/hdivnstsimple/veclaplace.html b/MPIparallel/hdivnstsimple/veclaplace.html index 66322a81..ab97ec91 100644 --- a/MPIparallel/hdivnstsimple/veclaplace.html +++ b/MPIparallel/hdivnstsimple/veclaplace.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@ @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -576,7 +608,7 @@

    Contents

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/MPIparallel/intro.html b/MPIparallel/intro.html index c26b4a02..45aa0505 100644 --- a/MPIparallel/intro.html +++ b/MPIparallel/intro.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@ @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -509,7 +541,7 @@

    The MPI library - © Copyright 2023. + © Copyright 2024.

    diff --git a/MPIparallel/paralleliteration.html b/MPIparallel/paralleliteration.html index 0f9abcd7..9899ca7f 100644 --- a/MPIparallel/paralleliteration.html +++ b/MPIparallel/paralleliteration.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@     @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -627,7 +659,7 @@

    Richardson iteration - © Copyright 2023. + © Copyright 2024.

    diff --git a/MPIparallel/parallelstokes.html b/MPIparallel/parallelstokes.html index ac588ed1..9a6d50f2 100644 --- a/MPIparallel/parallelstokes.html +++ b/MPIparallel/parallelstokes.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@     @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -517,7 +549,7 @@

    Solving Stokes in parallel - © Copyright 2023. + © Copyright 2024.

    diff --git a/MPIparallel/petsc.html b/MPIparallel/petsc.html index 6404fdfe..d91e2ec4 100644 --- a/MPIparallel/petsc.html +++ b/MPIparallel/petsc.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@     @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -551,7 +583,7 @@

    Using PETSc - © Copyright 2023. + © Copyright 2024.

    diff --git a/MPIparallel/vectypes.html b/MPIparallel/vectypes.html index 506b0f2b..2e1621d7 100644 --- a/MPIparallel/vectypes.html +++ b/MPIparallel/vectypes.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -722,7 +754,7 @@

    Vector operations: - © Copyright 2023. + © Copyright 2024.

    diff --git a/README.html b/README.html index d7db86a2..243bc39a 100644 --- a/README.html +++ b/README.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@     @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -406,7 +438,7 @@

    This repository contains an interactive introduction to the Finite Element M

    - © Copyright 2023. + © Copyright 2024.

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b/_sources/DG/fourthorder.ipynb index a089d70b..5ca1bbc9 100644 --- a/_sources/DG/fourthorder.ipynb +++ b/_sources/DG/fourthorder.ipynb @@ -43,19 +43,19 @@ }, { "cell_type": "code", - "execution_count": 1, + "execution_count": null, "metadata": {}, "outputs": [], "source": [ - "import netgen.gui\n", "from ngsolve import *\n", + "from ngsolve.webgui import Draw\n", "from netgen.geom2d import unit_square\n", "mesh = Mesh (unit_square.GenerateMesh(maxh=0.1))" ] }, { "cell_type": "code", - "execution_count": 12, + "execution_count": null, "metadata": {}, "outputs": [], "source": [ @@ -79,7 +79,7 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": null, "metadata": {}, "outputs": [], "source": [ @@ -107,7 +107,7 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": null, "metadata": {}, "outputs": [], "source": [ @@ -129,7 +129,7 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3", + "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, @@ -143,9 +143,9 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.7.2" + "version": "3.12.2" } }, "nbformat": 4, - "nbformat_minor": 2 + "nbformat_minor": 4 } diff --git a/_sources/iFEM.ipynb b/_sources/iFEM.ipynb index 837a6de8..d1f00dfc 100644 --- a/_sources/iFEM.ipynb +++ b/_sources/iFEM.ipynb @@ -280,7 +280,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.12.1" + "version": "3.12.2" } }, "nbformat": 4, diff --git a/_sources/intro.md b/_sources/intro.md index cefd852f..b56776af 100644 --- a/_sources/intro.md +++ b/_sources/intro.md @@ -8,12 +8,9 @@ TU Wien, [Institute of Analysis and Scientific Computing](https://www.asc.tuwien ... -Many thanks go to [Matthias Hochsteger](https://www.linkedin.com/in/matthias-hochsteger-316213196) and [Christopher Lackner](https://www.linkedin.com/in/christopher-lackner-2ab075191) for their technical help, and to [Edoardo Bonetti](https://www.asc.tuwien.ac.at/?id=scicomp/people) for careful checking. -This lecture is designed for students in applied mathematics ... -in an early stage. -It is given in this form the first time in winter term 23/24. +This lecture is given in this form the first time in summer term 24. If you have suggestions for improvements, or found some errors, please send them per mail to the author, or open a pull request on the source repo of the book. Many section are still in draft version, and will be cleaned as the class proceeds. diff --git a/domaindecomposition/BDDC.html b/domaindecomposition/BDDC.html index dcc5cf10..30094634 100644 --- a/domaindecomposition/BDDC.html +++ b/domaindecomposition/BDDC.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@ @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -527,7 +559,7 @@

    BDDC - Preconditioner - © Copyright 2023. + © Copyright 2024.

    diff --git a/domaindecomposition/FETI-DP.html b/domaindecomposition/FETI-DP.html index 926eb2d0..a96e715a 100644 --- a/domaindecomposition/FETI-DP.html +++ b/domaindecomposition/FETI-DP.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -579,7 +611,7 @@

    FETI-DP#<

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/domaindecomposition/FETI.html b/domaindecomposition/FETI.html index c1341e05..29ee7528 100644 --- a/domaindecomposition/FETI.html +++ b/domaindecomposition/FETI.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -643,7 +675,7 @@

    Preconditioner for \(S\) - © Copyright 2023. + © Copyright 2024.

    diff --git a/domaindecomposition/introduction.html b/domaindecomposition/introduction.html index af1d4d04..4f462cd3 100644 --- a/domaindecomposition/introduction.html +++ b/domaindecomposition/introduction.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -447,7 +479,7 @@

    Domain Decomposition with Lagrange parameters - © Copyright 2023. + © Copyright 2024.

    diff --git a/domaindecomposition/traces.html b/domaindecomposition/traces.html index 51bec4ce..12dc77dc 100644 --- a/domaindecomposition/traces.html +++ b/domaindecomposition/traces.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -544,7 +576,7 @@

    Trace norm on boundary sub-domains - © Copyright 2023. + © Copyright 2024.

    diff --git a/ex20210609.html b/ex20210609.html index 6609d320..0cee7358 100644 --- a/ex20210609.html +++ b/ex20210609.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -650,7 +682,7 @@

    Implement a parallel BDDC preconditioner - © Copyright 2023. + © Copyright 2024.

    diff --git a/genindex.html b/genindex.html index 18029442..12a19585 100644 --- a/genindex.html +++ b/genindex.html @@ -181,6 +181,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -329,7 +361,7 @@

    Index

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/helmholtz/absorbing.html b/helmholtz/absorbing.html index a01e62c5..340d13a8 100644 --- a/helmholtz/absorbing.html +++ b/helmholtz/absorbing.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@ @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -482,7 +514,7 @@

    Helmholtz Equation - © Copyright 2023. + © Copyright 2024.

    diff --git a/helmholtz/grating.html b/helmholtz/grating.html index 4f58b35a..780d4ba6 100644 --- a/helmholtz/grating.html +++ b/helmholtz/grating.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@     @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -460,7 +492,7 @@

    Grating#<

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/iFEM.html b/iFEM.html index d353c731..6ee63ba6 100644 --- a/iFEM.html +++ b/iFEM.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -675,7 +707,7 @@

    Some inequalities in Sobolev spaces - © Copyright 2023. + © Copyright 2024.

    diff --git a/intro.html b/intro.html index 99daa825..ab672679 100644 --- a/intro.html +++ b/intro.html @@ -183,6 +183,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -338,10 +370,7 @@

    An Interactive Introduction to the Finite Element MethodJoachim Schöberl

    TU Wien, Institute of Analysis and Scientific Computing

    …

    -

    Many thanks go to Matthias Hochsteger and Christopher Lackner for their technical help, and to Edoardo Bonetti for careful checking.

    -

    This lecture is designed for students in applied mathematics … -in an early stage. -It is given in this form the first time in winter term 23/24. +

    This lecture is given in this form the first time in summer term 24. If you have suggestions for improvements, or found some errors, please send them per mail to the author, or open a pull request on the source repo of the book. Many section are still in draft version, and will be cleaned as the class proceeds.

    @@ -354,6 +383,55 @@

    An Interactive Introduction to the Finite Element Method4. Iterative Solvers +
    +

    Mixed Finite Element Methods

    + +
    +
    +

    Discontinuous Galerkin Methods

    + +
    +
    +

    Mixed Methods for Second Order Equations

    + +
    +
    +

    Iteration Methods

    + +

    + + + + + @@ -123,8 +128,6 @@ - -
    @@ -174,7 +177,7 @@ @@ -184,6 +187,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -332,8 +367,8 @@

    Contents

    @@ -346,7 +381,7 @@

    Contents

    -

    The Chebyshev Method#

    +

    26. The Chebyshev Method#

    Let \(A\) be SPD, and \(C\) an SPD preconditioner. If we perform \(n\) steps of the Richardson iteration with damping parameter \(\tau\), the error \(e^n\) follows from the initial error \(e^0\) via

    \[ @@ -385,7 +420,7 @@

    The Chebyshev Method

    The solution to this min-max problem is given by Chebyshev polynomials.

    -

    Chebyshev polynomials#

    +

    26.1. Chebyshev polynomials#

    Chebyshev polynomials (of the first kind) are defined via the three-term recurrence relation

    \begin{eqnarray*} T_0(x) & = & 1 \ @@ -435,6 +470,11 @@

    Chebyshev polynomials

    +
    +
    <matplotlib.legend.Legend at 0x10b946090>
+
    +
    +

    We rescale

    -

    The Chebyshev iteration#

    +

    26.2. The Chebyshev iteration#

    If we choose the optimal damping parameters \(\tau_i\) for \(n\) steps we obtain an error

    \[ @@ -653,6 +698,11 @@

    The Chebyshev iteration

    +
    +
    0.023921286926102105 1.6757907507089311
+
    +
    +
    @@ -660,6 +710,89 @@

    The Chebyshev iteration

    +
    +
    Iteration 0 err= 0.01582110896407892
+Iteration 1 err= 0.013922181481556276
+Iteration 2 err= 0.011476030492532669
+Iteration 3 err= 0.009059023137271987
+Iteration 4 err= 0.007052356124173123
+Iteration 5 err= 0.005578858264906384
+Iteration 6 err= 0.004587596529114102
+Iteration 7 err= 0.0038718847011865016
+Iteration 8 err= 0.0032945237386851466
+Iteration 9 err= 0.002751549230743565
+Iteration 10 err= 0.0022497739915375917
+Iteration 11 err= 0.001803593976333938
+Iteration 12 err= 0.0014128271916008758
+Iteration 13 err= 0.0010856674159789273
+Iteration 14 err= 0.0008202093227311343
+Iteration 15 err= 0.0006155748247353796
+Iteration 16 err= 0.0004613688818919865
+Iteration 17 err= 0.0003507549799964684
+Iteration 18 err= 0.0002728005235373108
+Iteration 19 err= 0.00021595079353913196
+Iteration 20 err= 0.00017570605885732808
+Iteration 21 err= 0.00014492153597237955
+Iteration 22 err= 0.00011998169981225712
+Iteration 23 err= 9.848070283245423e-05
+Iteration 24 err= 7.957767469477019e-05
+Iteration 25 err= 6.306298982519902e-05
+Iteration 26 err= 4.868841925515925e-05
+Iteration 27 err= 3.682599162697037e-05
+Iteration 28 err= 2.7259399075533895e-05
+Iteration 29 err= 2.0046442873567982e-05
+Iteration 30 err= 1.4888122144003478e-05
+Iteration 31 err= 1.1454113319650339e-05
+Iteration 32 err= 9.257788714659064e-06
+Iteration 33 err= 7.76628215878487e-06
+Iteration 34 err= 6.588857542533488e-06
+Iteration 35 err= 5.513610076479524e-06
+Iteration 36 err= 4.516898982393652e-06
+Iteration 37 err= 3.597623279322541e-06
+Iteration 38 err= 2.7815384689105085e-06
+Iteration 39 err= 2.0965434266789667e-06
+Iteration 40 err= 1.5439796579189577e-06
+Iteration 41 err= 1.1275880795336662e-06
+Iteration 42 err= 8.385618400239801e-07
+Iteration 43 err= 6.495797078809779e-07
+Iteration 44 err= 5.248690304291037e-07
+Iteration 45 err= 4.3525124728432896e-07
+Iteration 46 err= 3.63730727663832e-07
+Iteration 47 err= 3.0175216009825165e-07
+Iteration 48 err= 2.465303970244925e-07
+Iteration 49 err= 1.9662103485504439e-07
+Iteration 50 err= 1.5343034407815853e-07
+Iteration 51 err= 1.1755537957497641e-07
+Iteration 52 err= 8.935453489003239e-08
+Iteration 53 err= 6.729788755286304e-08
+Iteration 54 err= 5.064503053238712e-08
+Iteration 55 err= 3.8306100788362255e-08
+Iteration 56 err= 2.9714057523794218e-08
+Iteration 57 err= 2.3707713672918417e-08
+Iteration 58 err= 1.9266656449020162e-08
+Iteration 59 err= 1.5854114285166374e-08
+Iteration 60 err= 1.3059491673335996e-08
+Iteration 61 err= 1.0735847431651127e-08
+Iteration 62 err= 8.686494452899965e-09
+Iteration 63 err= 6.867659846468332e-09
+Iteration 64 err= 5.302347984388261e-09
+Iteration 65 err= 3.990474165126989e-09
+Iteration 66 err= 2.957026375736346e-09
+Iteration 67 err= 2.1703862973201954e-09
+Iteration 68 err= 1.616153422513513e-09
+Iteration 69 err= 1.250889053109186e-09
+Iteration 70 err= 1.0177157679035385e-09
+Iteration 71 err= 8.549287598746731e-10
+Iteration 72 err= 7.252273880727779e-10
+Iteration 73 err= 6.069788104427646e-10
+Iteration 74 err= 4.943582107671246e-10
+Iteration 75 err= 3.914858880356967e-10
+Iteration 76 err= 3.0075334036272057e-10
+Iteration 77 err= 2.2568072867770525e-10
+Iteration 78 err= 1.6613585858432592e-10
+
    +
    +

    Experiments:

    @@ -708,6 +846,24 @@

    The Chebyshev iteration
    + + +
    +

    previous

    +

    24. Preconditioning

    +
    +     + +
    +

    next

    +

    27. Conjugate Gradients

    +
    + +
    @@ -724,8 +880,8 @@

    The Chebyshev iteration @@ -750,7 +906,7 @@

    The Chebyshev iteration - © Copyright 2023. + © Copyright 2024.

    diff --git a/iterative/Richardson.html b/iterative/Richardson.html index c28436dc..aa875d91 100644 --- a/iterative/Richardson.html +++ b/iterative/Richardson.html @@ -8,7 +8,7 @@ - The Richardson Iteration — Introduction to Scientific Computing + 20. The Richardson Iteration — Introduction to Scientific Computing @@ -32,7 +32,7 @@ - + @@ -59,9 +59,14 @@ + + + + + @@ -123,8 +128,6 @@ - -
    @@ -174,7 +177,7 @@ @@ -184,6 +187,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -332,9 +367,9 @@

    Contents

    @@ -348,7 +383,7 @@

    Contents

    -

    The Richardson Iteration#

    +

    20. The Richardson Iteration#

    also called simple iteration is the fixed point iteration

    \[ @@ -419,7 +454,7 @@

    The Richardson Iteration -

    Optimizing the relaxation parameter \(\alpha\)#

    +

    21. Optimizing the relaxation parameter \(\alpha\)#

    Let \(A\) be SPD, and let \(\sigma(A) = \{ \lambda_i \in {\mathbb R} \}\) with \(0 < \lambda_1 \leq \lambda_2 \ldots \leq \lambda_n\).

    Then the eigenvalues of the error propagation matrix \(M = I - \alpha A\) are \(\{ 1 - \alpha \lambda_i \}\).

    Whenever we choose @@ -458,7 +493,7 @@

    Optimizing the relaxation parameter -

    Experiments with the Richardson iteration#

    +

    22. Experiments with the Richardson iteration#

    from ngsolve import *
@@ -494,6 +529,30 @@ 
    Experiments with the Richardson iteration
+
    1.650595133550197
+4.581356277192373
+4.878844983392103
+5.01336629024651
+5.105014615250208
+5.176320540449654
+5.234675552913327
+5.283628208207171
+5.325303075350094
+5.361132400705149
+5.392152380486606
+5.419150349229767
+5.442747024022966
+5.463445615167365
+5.481662515543909
+5.497747273611648
+5.511996158845851
+5.5246618094460285
+5.535960424218709
+5.546077363788045
+
    
+
    
+
    @@ -513,6 +572,1396 @@

    Experiments with the Richardson iteration +
    iteration 0 res= 0.02948894632867606
+iteration 1 res= 0.028218356512525144
+iteration 2 res= 0.027404731964668753
+iteration 3 res= 0.026739064015994258
+iteration 4 res= 0.026149886985713675
+iteration 5 res= 0.025607091174332675
+iteration 6 res= 0.025095891354368766
+iteration 7 res= 0.024608338907413877
+iteration 8 res= 0.02413981680402453
+iteration 9 res= 0.023687443874675685
+iteration 10 res= 0.023249298406432254
+iteration 11 res= 0.022824018558386856
+iteration 12 res= 0.022410586053865696
+iteration 13 res= 0.02200820353681429
+iteration 14 res= 0.02161622200075671
+iteration 15 res= 0.021234096118907843
+iteration 16 res= 0.020861355728149778
+iteration 17 res= 0.020497587003014767
+iteration 18 res= 0.02014241963601273
+iteration 19 res= 0.019795517855411247
+iteration 20 res= 0.019456573964103333
+iteration 21 res= 0.01912530357778844
+iteration 22 res= 0.01880144203589283
+iteration 23 res= 0.018484741639591858
+iteration 24 res= 0.018174969484979445
+iteration 25 res= 0.017871905732521804
+iteration 26 res= 0.01757534220193772
+iteration 27 res= 0.01728508121380277
+iteration 28 res= 0.017000934621104014
+iteration 29 res= 0.016722722989178505
+iteration 30 res= 0.0164502748931743
+iteration 31 res= 0.01618342630981679
+iteration 32 res= 0.0159220200857929
+iteration 33 res= 0.015665905469114498
+iteration 34 res= 0.015414937692821994
+iteration 35 res= 0.015168977602635123
+iteration 36 res= 0.014927891321857916
+iteration 37 res= 0.014691549948144927
+iteration 38 res= 0.014459829277739242
+iteration 39 res= 0.014232609553575528
+iteration 40 res= 0.01400977523425765
+iteration 41 res= 0.013791214781409204
+iteration 42 res= 0.013576820463288576
+iteration 43 res= 0.013366488172876428
+iteration 44 res= 0.013160117258903219
+iteration 45 res= 0.012957610368496578
+iteration 46 res= 0.012758873300305064
+iteration 47 res= 0.012563814867101952
+iteration 48 res= 0.012372346766996847
+iteration 49 res= 0.012184383462487215
+iteration 50 res= 0.011999842066672047
+iteration 51 res= 0.011818642236025545
+iteration 52 res= 0.011640706069195395
+iteration 53 res= 0.011465958011346781
+iteration 54 res= 0.011294324763623213
+iteration 55 res= 0.011125735197338492
+iteration 56 res= 0.01096012027255244
+iteration 57 res= 0.010797412960715985
+iteration 58 res= 0.010637548171101649
+iteration 59 res= 0.010480462680760576
+iteration 60 res= 0.01032609506777182
+iteration 61 res= 0.010174385647569131
+iteration 62 res= 0.010025276412150008
+iteration 63 res= 0.009878710971987762
+iteration 64 res= 0.009734634500482446
+iteration 65 res= 0.009592993680800075
+iteration 66 res= 0.009453736654961228
+iteration 67 res= 0.009316812975051212
+iteration 68 res= 0.009182173556434062
+iteration 69 res= 0.009049770632860905
+iteration 70 res= 0.00891955771337195
+iteration 71 res= 0.008791489540898312
+iteration 72 res= 0.008665522052476634
+iteration 73 res= 0.00854161234099557
+iteration 74 res= 0.008419718618398703
+iteration 75 res= 0.008299800180273454
+iteration 76 res= 0.008181817371760405
+iteration 77 res= 0.008065731554721322
+iteration 78 res= 0.00795150507610824
+iteration 79 res= 0.007839101237479745
+iteration 80 res= 0.007728484265613289
+iteration 81 res= 0.007619619284166102
+iteration 82 res= 0.00751247228633936
+iteration 83 res= 0.007407010108503324
+iteration 84 res= 0.007303200404743258
+iteration 85 res= 0.007201011622288332
+iteration 86 res= 0.007100412977787459
+iteration 87 res= 0.007001374434398191
+iteration 88 res= 0.006903866679656398
+iteration 89 res= 0.006807861104096061
+iteration 90 res= 0.006713329780590161
+iteration 91 res= 0.006620245444384857
+iteration 92 res= 0.006528581473800669
+iteration 93 res= 0.006438311871575546
+iteration 94 res= 0.006349411246825661
+iteration 95 res= 0.006261854797601374
+iteration 96 res= 0.006175618294016112
+iteration 97 res= 0.006090678061927554
+iteration 98 res= 0.006007010967150938
+iteration 99 res= 0.005924594400185357
+iteration 100 res= 0.005843406261434845
+iteration 101 res= 0.005763424946906339
+iteration 102 res= 0.005684629334367998
+iteration 103 res= 0.00560699876995136
+iteration 104 res= 0.005530513055182119
+iteration 105 res= 0.0054551524344242115
+iteration 106 res= 0.00538089758272323
+iteration 107 res= 0.00530772959403499
+iteration 108 res= 0.005235629969826383
+iteration 109 res= 0.005164580608035319
+iteration 110 res= 0.005094563792377851
+iteration 111 res= 0.005025562181990357
+iteration 112 res= 0.004957558801395561
+iteration 113 res= 0.004890537030781347
+iteration 114 res= 0.004824480596581759
+iteration 115 res= 0.004759373562350154
+iteration 116 res= 0.004695200319914352
+iteration 117 res= 0.00463194558080472
+iteration 118 res= 0.00456959436794578
+iteration 119 res= 0.004508132007602535
+iteration 120 res= 0.004447544121573191
+iteration 121 res= 0.004387816619619899
+iteration 122 res= 0.004328935692129654
+iteration 123 res= 0.004270887802997769
+iteration 124 res= 0.004213659682726487
+iteration 125 res= 0.004157238321731646
+iteration 126 res= 0.004101610963850527
+iteration 127 res= 0.004046765100044237
+iteration 128 res= 0.003992688462288246
+iteration 129 res= 0.003939369017644903
+iteration 130 res= 0.0038867949625118877
+iteration 131 res= 0.0038349547170410097
+iteration 132 res= 0.0037838369197214696
+iteration 133 res= 0.003733430422122664
+iteration 134 res= 0.0036837242837908417
+iteration 135 res= 0.0036347077672949186
+iteration 136 res= 0.0035863703334165784
+iteration 137 res= 0.003538701636479851
+iteration 138 res= 0.0034916915198156353
+iteration 139 res= 0.003445330011357056
+iteration 140 res= 0.003399607319361085
+iteration 141 res= 0.0033545138282525524
+iteration 142 res= 0.0033100400945866196
+iteration 143 res= 0.0032661768431257593
+iteration 144 res= 0.0032229149630277794
+iteration 145 res= 0.003180245504141052
+iteration 146 res= 0.0031381596734038537
+iteration 147 res= 0.003096648831344128
+iteration 148 res= 0.003055704488676831
+iteration 149 res= 0.0030153183029954264
+iteration 150 res= 0.0029754820755548473
+iteration 151 res= 0.002936187748142723
+iteration 152 res= 0.0028974274000363328
+iteration 153 res= 0.0028591932450424094
+iteration 154 res= 0.002821477628617289
+iteration 155 res= 0.0027842730250648108
+iteration 156 res= 0.0027475720348096233
+iteration 157 res= 0.002711367381743343
+iteration 158 res= 0.002675651910641553
+iteration 159 res= 0.0026404185846491523
+iteration 160 res= 0.0026056604828323076
+iteration 161 res= 0.002571370797794421
+iteration 162 res= 0.0025375428333546724
+iteration 163 res= 0.0025041700022867723
+iteration 164 res= 0.0024712458241164356
+iteration 165 res= 0.0024387639229754535
+iteration 166 res= 0.002406718025510794
+iteration 167 res= 0.0023751019588471186
+iteration 168 res= 0.0023439096486008943
+iteration 169 res= 0.002313135116944736
+iteration 170 res= 0.002282772480720254
+iteration 171 res= 0.0022528159495981805
+iteration 172 res= 0.002223259824284108
+iteration 173 res= 0.0021940984947686703
+iteration 174 res= 0.0021653264386206975
+iteration 175 res= 0.002136938219322149
+iteration 176 res= 0.0021089284846435748
+iteration 177 res= 0.0020812919650588426
+iteration 178 res= 0.002054023472198047
+iteration 179 res= 0.0020271178973373977
+iteration 180 res= 0.002000570209925151
+iteration 181 res= 0.001974375456142303
+iteration 182 res= 0.0019485287574972747
+iteration 183 res= 0.0019230253094533658
+iteration 184 res= 0.0018978603800883275
+iteration 185 res= 0.001873029308784762
+iteration 186 res= 0.0018485275049508396
+iteration 187 res= 0.0018243504467700994
+iteration 188 res= 0.0018004936799799042
+iteration 189 res= 0.0017769528166773547
+iteration 190 res= 0.0017537235341521322
+iteration 191 res= 0.0017308015737454805
+iteration 192 res= 0.0017081827397344514
+iteration 193 res= 0.0016858628982409185
+iteration 194 res= 0.0016638379761645312
+iteration 195 res= 0.0016421039601389302
+iteration 196 res= 0.0016206568955107815
+iteration 197 res= 0.001599492885340684
+iteration 198 res= 0.0015786080894256611
+iteration 199 res= 0.0015579987233424746
+iteration 200 res= 0.0015376610575112638
+iteration 201 res= 0.0015175914162788996
+iteration 202 res= 0.001497786177021638
+iteration 203 res= 0.0014782417692664356
+iteration 204 res= 0.0014589546738305028
+iteration 205 res= 0.0014399214219786639
+iteration 206 res= 0.001421138594597882
+iteration 207 res= 0.0014026028213886415
+iteration 208 res= 0.0013843107800728402
+iteration 209 res= 0.0013662591956174583
+iteration 210 res= 0.0013484448394739108
+iteration 211 res= 0.0013308645288324717
+iteration 212 res= 0.001313515125891606
+iteration 213 res= 0.0012963935371414679
+iteration 214 res= 0.001279496712661682
+iteration 215 res= 0.0012628216454326235
+iteration 216 res= 0.001246365370660102
+iteration 217 res= 0.0012301249651130908
+iteration 218 res= 0.0012140975464740651
+iteration 219 res= 0.0011982802727017908
+iteration 220 res= 0.0011826703414061588
+iteration 221 res= 0.0011672649892346984
+iteration 222 res= 0.0011520614912708586
+iteration 223 res= 0.0011370571604431875
+iteration 224 res= 0.0011222493469456843
+iteration 225 res= 0.0011076354376687463
+iteration 226 res= 0.0010932128556406333
+iteration 227 res= 0.0010789790594789973
+iteration 228 res= 0.001064931542852534
+iteration 229 res= 0.0010510678339522277
+iteration 230 res= 0.001037385494972173
+iteration 231 res= 0.0010238821215996488
+iteration 232 res= 0.0010105553425142871
+iteration 233 res= 0.0009974028188960637
+iteration 234 res= 0.0009844222439420404
+iteration 235 res= 0.0009716113423914745
+iteration 236 res= 0.0009589678700593078
+iteration 237 res= 0.0009464896133776843
+iteration 238 res= 0.000934174388945475
+iteration 239 res= 0.0009220200430853675
+iteration 240 res= 0.0009100244514088579
+iteration 241 res= 0.0008981855183882589
+iteration 242 res= 0.0008865011769363411
+iteration 243 res= 0.000874969387992818
+iteration 244 res= 0.0008635881401178498
+iteration 245 res= 0.0008523554490924386
+iteration 246 res= 0.0008412693575253815
+iteration 247 res= 0.0008303279344667386
+iteration 248 res= 0.0008195292750277786
+iteration 249 res= 0.0008088715000070816
+iteration 250 res= 0.0007983527555228889
+iteration 251 res= 0.0007879712126512157
+iteration 252 res= 0.0007777250670701908
+iteration 253 res= 0.0007676125387098062
+iteration 254 res= 0.0007576318714075688
+iteration 255 res= 0.0007477813325695091
+iteration 256 res= 0.0007380592128367267
+iteration 257 res= 0.0007284638257572724
+iteration 258 res= 0.0007189935074629959
+iteration 259 res= 0.0007096466163518835
+iteration 260 res= 0.0007004215327750499
+iteration 261 res= 0.0006913166587289043
+iteration 262 res= 0.0006823304175519718
+iteration 263 res= 0.0006734612536264959
+iteration 264 res= 0.0006647076320847242
+iteration 265 res= 0.0006560680385196472
+iteration 266 res= 0.0006475409787002921
+iteration 267 res= 0.0006391249782913056
+iteration 268 res= 0.0006308185825769721
+iteration 269 res= 0.0006226203561892683
+iteration 270 res= 0.0006145288828402268
+iteration 271 res= 0.0006065427650583211
+iteration 272 res= 0.0005986606239286902
+iteration 273 res= 0.0005908810988375774
+iteration 274 res= 0.0005832028472203714
+iteration 275 res= 0.0005756245443134728
+iteration 276 res= 0.0005681448829100217
+iteration 277 res= 0.0005607625731190868
+iteration 278 res= 0.0005534763421285083
+iteration 279 res= 0.0005462849339713222
+iteration 280 res= 0.0005391871092955329
+iteration 281 res= 0.0005321816451374198
+iteration 282 res= 0.0005252673346979963
+iteration 283 res= 0.0005184429871229609
+iteration 284 res= 0.0005117074272857038
+iteration 285 res= 0.0005050594955735753
+iteration 286 res= 0.0004984980476772953
+iteration 287 res= 0.0004920219543833185
+iteration 288 res= 0.0004856301013692845
+iteration 289 res= 0.0004793213890024514
+iteration 290 res= 0.0004730947321409969
+iteration 291 res= 0.0004669490599382262
+iteration 292 res= 0.00046088331564949606
+iteration 293 res= 0.00045489645644206594
+iteration 294 res= 0.0004489874532075736
+iteration 295 res= 0.0004431552903771559
+iteration 296 res= 0.0004373989657393343
+iteration 297 res= 0.00043171749026038863
+iteration 298 res= 0.0004261098879073096
+iteration 299 res= 0.0004205751954732711
+iteration 300 res= 0.00041511246240560586
+iteration 301 res= 0.00040972075063613406
+iteration 302 res= 0.0004043991344139413
+iteration 303 res= 0.0003991467001405369
+iteration 304 res= 0.00039396254620725773
+iteration 305 res= 0.0003888457828349773
+iteration 306 res= 0.00038379553191612067
+iteration 307 res= 0.0003788109268588321
+iteration 308 res= 0.00037389111243330426
+iteration 309 res= 0.00036903524462027096
+iteration 310 res= 0.00036424249046166534
+iteration 311 res= 0.00035951202791324575
+iteration 312 res= 0.0003548430456994067
+iteration 313 res= 0.00035023474316978965
+iteration 314 res= 0.00034568633015818925
+iteration 315 res= 0.0003411970268430543
+iteration 316 res= 0.00033676606361021915
+iteration 317 res= 0.00033239268091735845
+iteration 318 res= 0.0003280761291603079
+iteration 319 res= 0.00032381566854135425
+iteration 320 res= 0.00031961056893919514
+iteration 321 res= 0.00031546010978072525
+iteration 322 res= 0.00031136357991460727
+iteration 323 res= 0.00030732027748656425
+iteration 324 res= 0.00030332950981633533
+iteration 325 res= 0.0002993905932763787
+iteration 326 res= 0.00029550285317220335
+iteration 327 res= 0.00029166562362421806
+iteration 328 res= 0.0002878782474514669
+iteration 329 res= 0.00028414007605657705
+iteration 330 res= 0.0002804504693126494
+iteration 331 res= 0.000276808795451333
+iteration 332 res= 0.00027321443095271923
+iteration 333 res= 0.0002696667604364646
+iteration 334 res= 0.0002661651765546381
+iteration 335 res= 0.00026270907988576106
+iteration 336 res= 0.00025929787883052676
+iteration 337 res= 0.00025593098950869945
+iteration 338 res= 0.0002526078356576112
+iteration 339 res= 0.00024932784853185495
+iteration 340 res= 0.0002460904668044248
+iteration 341 res= 0.0002428951364691859
+iteration 342 res= 0.00023974131074455462
+iteration 343 res= 0.00023662844997865775
+iteration 344 res= 0.00023355602155555058
+iteration 345 res= 0.000230523499802865
+iteration 346 res= 0.00022753036590056257
+iteration 347 res= 0.00022457610779107428
+iteration 348 res= 0.00022166022009041344
+iteration 349 res= 0.00021878220400070643
+iteration 350 res= 0.00021594156722375184
+iteration 351 res= 0.00021313782387576326
+iteration 352 res= 0.00021037049440326085
+iteration 353 res= 0.00020763910550014816
+iteration 354 res= 0.0002049431900257017
+iteration 355 res= 0.0002022822869238887
+iteration 356 res= 0.0001996559411436378
+iteration 357 res= 0.00019706370356012813
+iteration 358 res= 0.00019450513089724127
+iteration 359 res= 0.00019197978565093756
+iteration 360 res= 0.00018948723601375087
+iteration 361 res= 0.00018702705580014213
+iteration 362 res= 0.00018459882437305076
+iteration 363 res= 0.0001822021265712087
+iteration 364 res= 0.0001798365526375558
+iteration 365 res= 0.00017750169814854242
+iteration 366 res= 0.00017519716394439477
+iteration 367 res= 0.0001729225560603059
+iteration 368 res= 0.00017067748565849998
+iteration 369 res= 0.00016846156896120233
+iteration 370 res= 0.00016627442718465936
+iteration 371 res= 0.00016411568647362458
+iteration 372 res= 0.0001619849778372474
+iteration 373 res= 0.0001598819370853511
+iteration 374 res= 0.00015780620476577115
+iteration 375 res= 0.00015575742610255484
+iteration 376 res= 0.00015373525093482907
+iteration 377 res= 0.00015173933365657038
+iteration 378 res= 0.00014976933315716793
+iteration 379 res= 0.0001478249127627026
+iteration 380 res= 0.00014590574017811026
+iteration 381 res= 0.00014401148742999368
+iteration 382 res= 0.00014214183081026426
+iteration 383 res= 0.00014029645082048867
+iteration 384 res= 0.00013847503211698154
+iteration 385 res= 0.00013667726345660295
+iteration 386 res= 0.00013490283764327728
+iteration 387 res= 0.00013315145147529996
+iteration 388 res= 0.00013142280569310838
+iteration 389 res= 0.00012971660492803406
+iteration 390 res= 0.00012803255765146782
+iteration 391 res= 0.0001263703761248743
+iteration 392 res= 0.00012472977635038387
+iteration 393 res= 0.00012311047802202467
+iteration 394 res= 0.0001215122044776023
+iteration 395 res= 0.00011993468265125935
+iteration 396 res= 0.00011837764302659063
+iteration 397 res= 0.00011684081959040211
+iteration 398 res= 0.00011532394978709451
+iteration 399 res= 0.00011382677447359922
+iteration 400 res= 0.00011234903787495147
+iteration 401 res= 0.00011089048754040256
+iteration 402 res= 0.00010945087430017316
+iteration 403 res= 0.00010802995222266234
+iteration 404 res= 0.00010662747857234237
+iteration 405 res= 0.00010524321376811381
+iteration 406 res= 0.00010387692134226174
+iteration 407 res= 0.00010252836789988472
+iteration 408 res= 0.00010119732307894057
+iteration 409 res= 9.988355951071648e-05
+iteration 410 res= 9.858685278094369e-05
+iteration 411 res= 9.730698139124936e-05
+iteration 412 res= 9.604372672130701e-05
+iteration 413 res= 9.47968729912945e-05
+iteration 414 res= 9.356620722500623e-05
+iteration 415 res= 9.235151921329459e-05
+iteration 416 res= 9.115260147814261e-05
+iteration 417 res= 8.996924923707113e-05
+iteration 418 res= 8.880126036811344e-05
+iteration 419 res= 8.764843537518527e-05
+iteration 420 res= 8.65105773539008e-05
+iteration 421 res= 8.53874919578967e-05
+iteration 422 res= 8.427898736555464e-05
+iteration 423 res= 8.318487424713558e-05
+iteration 424 res= 8.210496573238421e-05
+iteration 425 res= 8.10390773785428e-05
+iteration 426 res= 7.998702713878978e-05
+iteration 427 res= 7.894863533108172e-05
+iteration 428 res= 7.7923724607356e-05
+iteration 429 res= 7.69121199232461e-05
+iteration 430 res= 7.591364850807186e-05
+iteration 431 res= 7.492813983531303e-05
+iteration 432 res= 7.3955425593414e-05
+iteration 433 res= 7.299533965700054e-05
+iteration 434 res= 7.204771805840642e-05
+iteration 435 res= 7.111239895970657e-05
+iteration 436 res= 7.018922262493387e-05
+iteration 437 res= 6.927803139285301e-05
+iteration 438 res= 6.83786696498858e-05
+iteration 439 res= 6.749098380357558e-05
+iteration 440 res= 6.661482225627753e-05
+iteration 441 res= 6.5750035379257e-05
+iteration 442 res= 6.48964754870327e-05
+iteration 443 res= 6.405399681218967e-05
+iteration 444 res= 6.322245548043824e-05
+iteration 445 res= 6.240170948597445e-05
+iteration 446 res= 6.159161866723811e-05
+iteration 447 res= 6.079204468289669e-05
+iteration 448 res= 6.000285098822809e-05
+iteration 449 res= 5.9223902811778785e-05
+iteration 450 res= 5.845506713226712e-05
+iteration 451 res= 5.769621265589978e-05
+iteration 452 res= 5.6947209793901234e-05
+iteration 453 res= 5.620793064033592e-05
+iteration 454 res= 5.547824895028533e-05
+iteration 455 res= 5.4758040118226835e-05
+iteration 456 res= 5.404718115676471e-05
+iteration 457 res= 5.3345550675539e-05
+iteration 458 res= 5.265302886058252e-05
+iteration 459 res= 5.1969497453756366e-05
+iteration 460 res= 5.129483973254997e-05
+iteration 461 res= 5.06289404901598e-05
+iteration 462 res= 4.997168601578292e-05
+iteration 463 res= 4.9322964075129095e-05
+iteration 464 res= 4.8682663891348016e-05
+iteration 465 res= 4.805067612597897e-05
+iteration 466 res= 4.742689286037963e-05
+iteration 467 res= 4.681120757716277e-05
+iteration 468 res= 4.620351514207797e-05
+iteration 469 res= 4.560371178602058e-05
+iteration 470 res= 4.501169508729309e-05
+iteration 471 res= 4.44273639540937e-05
+iteration 472 res= 4.3850618607254006e-05
+iteration 473 res= 4.3281360563201544e-05
+iteration 474 res= 4.2719492617101965e-05
+iteration 475 res= 4.216491882626763e-05
+iteration 476 res= 4.1617544493753685e-05
+iteration 477 res= 4.1077276152188614e-05
+iteration 478 res= 4.054402154778824e-05
+iteration 479 res= 4.001768962461957e-05
+iteration 480 res= 3.9498190509007414e-05
+iteration 481 res= 3.898543549418914e-05
+iteration 482 res= 3.847933702519887e-05
+iteration 483 res= 3.797980868383583e-05
+iteration 484 res= 3.7486765173977605e-05
+iteration 485 res= 3.7000122306936366e-05
+iteration 486 res= 3.6519796987111825e-05
+iteration 487 res= 3.6045707197830256e-05
+iteration 488 res= 3.55777719872434e-05
+iteration 489 res= 3.511591145457946e-05
+iteration 490 res= 3.4660046736434333e-05
+iteration 491 res= 3.4210099993364e-05
+iteration 492 res= 3.376599439653954e-05
+iteration 493 res= 3.332765411461013e-05
+iteration 494 res= 3.289500430079717e-05
+iteration 495 res= 3.246797108008226e-05
+iteration 496 res= 3.2046481536564425e-05
+iteration 497 res= 3.1630463701063014e-05
+iteration 498 res= 3.1219846538764815e-05
+iteration 499 res= 3.081455993713018e-05
+iteration 500 res= 3.0414534693897323e-05
+iteration 501 res= 3.0019702505262148e-05
+iteration 502 res= 2.962999595420444e-05
+iteration 503 res= 2.924534849901693e-05
+iteration 504 res= 2.8865694461862032e-05
+iteration 505 res= 2.8490969017638882e-05
+iteration 506 res= 2.8121108182833155e-05
+iteration 507 res= 2.7756048804650866e-05
+iteration 508 res= 2.739572855018196e-05
+iteration 509 res= 2.7040085895802202e-05
+iteration 510 res= 2.6689060116593705e-05
+iteration 511 res= 2.6342591276060492e-05
+iteration 512 res= 2.600062021585088e-05
+iteration 513 res= 2.566308854562293e-05
+iteration 514 res= 2.5329938633122654e-05
+iteration 515 res= 2.5001113594321307e-05
+iteration 516 res= 2.4676557283674938e-05
+iteration 517 res= 2.4356214284571195e-05
+iteration 518 res= 2.404002989985513e-05
+iteration 519 res= 2.3727950142494696e-05
+iteration 520 res= 2.3419921726297366e-05
+iteration 521 res= 2.3115892056957065e-05
+iteration 522 res= 2.2815809222863886e-05
+iteration 523 res= 2.2519621986457797e-05
+iteration 524 res= 2.222727977527399e-05
+iteration 525 res= 2.19387326734725e-05
+iteration 526 res= 2.1653931413235738e-05
+iteration 527 res= 2.1372827366316146e-05
+iteration 528 res= 2.109537253583884e-05
+iteration 529 res= 2.0821519548000633e-05
+iteration 530 res= 2.0551221644043878e-05
+iteration 531 res= 2.028443267225147e-05
+iteration 532 res= 2.0021107080061225e-05
+iteration 533 res= 1.9761199906272287e-05
+iteration 534 res= 1.9504666773424134e-05
+iteration 535 res= 1.925146388013875e-05
+iteration 536 res= 1.9001547993698304e-05
+iteration 537 res= 1.8754876442657965e-05
+iteration 538 res= 1.851140710949778e-05
+iteration 539 res= 1.8271098423537846e-05
+iteration 540 res= 1.80339093537432e-05
+iteration 541 res= 1.779979940174849e-05
+iteration 542 res= 1.756872859497817e-05
+iteration 543 res= 1.7340657479737663e-05
+iteration 544 res= 1.7115547114588773e-05
+iteration 545 res= 1.6893359063594896e-05
+iteration 546 res= 1.667405538980714e-05
+iteration 547 res= 1.6457598648798727e-05
+iteration 548 res= 1.6243951882208526e-05
+iteration 549 res= 1.603307861153314e-05
+iteration 550 res= 1.5824942831795766e-05
+iteration 551 res= 1.56195090054534e-05
+iteration 552 res= 1.541674205631676e-05
+iteration 553 res= 1.5216607363512638e-05
+iteration 554 res= 1.5019070755717276e-05
+iteration 555 res= 1.4824098505135998e-05
+iteration 556 res= 1.4631657321843358e-05
+iteration 557 res= 1.4441714348119533e-05
+iteration 558 res= 1.4254237152772588e-05
+iteration 559 res= 1.4069193725647556e-05
+iteration 560 res= 1.388655247213525e-05
+iteration 561 res= 1.3706282207796073e-05
+iteration 562 res= 1.352835215303197e-05
+iteration 563 res= 1.3352731927802536e-05
+iteration 564 res= 1.3179391546486105e-05
+iteration 565 res= 1.3008301412733248e-05
+iteration 566 res= 1.2839432314370949e-05
+iteration 567 res= 1.267275541853099e-05
+iteration 568 res= 1.2508242266573156e-05
+iteration 569 res= 1.2345864769364425e-05
+iteration 570 res= 1.2185595202389433e-05
+iteration 571 res= 1.202740620107054e-05
+iteration 572 res= 1.1871270756054911e-05
+iteration 573 res= 1.17171622086406e-05
+iteration 574 res= 1.1565054246186616e-05
+iteration 575 res= 1.141492089768125e-05
+iteration 576 res= 1.1266736529217028e-05
+iteration 577 res= 1.1120475839698992e-05
+iteration 578 res= 1.097611385644204e-05
+iteration 579 res= 1.0833625931010153e-05
+iteration 580 res= 1.0692987734899323e-05
+iteration 581 res= 1.0554175255490654e-05
+iteration 582 res= 1.0417164791840322e-05
+iteration 583 res= 1.0281932950723727e-05
+iteration 584 res= 1.014845664256488e-05
+iteration 585 res= 1.0016713077567441e-05
+iteration 586 res= 9.886679761795443e-06
+iteration 587 res= 9.758334493268848e-06
+iteration 588 res= 9.631655358306553e-06
+iteration 589 res= 9.50662072763806e-06
+iteration 590 res= 9.383209252835533e-06
+iteration 591 res= 9.261399862583206e-06
+iteration 592 res= 9.141171759091316e-06
+iteration 593 res= 9.022504414612773e-06
+iteration 594 res= 8.905377567832722e-06
+iteration 595 res= 8.789771220527565e-06
+iteration 596 res= 8.675665634014625e-06
+iteration 597 res= 8.563041325913588e-06
+iteration 598 res= 8.451879066724575e-06
+iteration 599 res= 8.342159876606776e-06
+iteration 600 res= 8.233865022079413e-06
+iteration 601 res= 8.126976012901917e-06
+iteration 602 res= 8.021474598829456e-06
+iteration 603 res= 7.917342766540326e-06
+iteration 604 res= 7.81456273657688e-06
+iteration 605 res= 7.713116960280898e-06
+iteration 606 res= 7.612988116816841e-06
+iteration 607 res= 7.514159110170325e-06
+iteration 608 res= 7.416613066315733e-06
+iteration 609 res= 7.320333330229059e-06
+iteration 610 res= 7.2253034631390215e-06
+iteration 611 res= 7.1315072396407e-06
+iteration 612 res= 7.038928645006803e-06
+iteration 613 res= 6.947551872370601e-06
+iteration 614 res= 6.857361320096953e-06
+iteration 615 res= 6.768341589047156e-06
+iteration 616 res= 6.680477480029778e-06
+iteration 617 res= 6.593753991136505e-06
+iteration 618 res= 6.5081563152482635e-06
+iteration 619 res= 6.42366983741779e-06
+iteration 620 res= 6.340280132459934e-06
+iteration 621 res= 6.257972962434929e-06
+iteration 622 res= 6.176734274248942e-06
+iteration 623 res= 6.096550197246109e-06
+iteration 624 res= 6.0174070408032425e-06
+iteration 625 res= 5.939291292072328e-06
+iteration 626 res= 5.862189613578814e-06
+iteration 627 res= 5.7860888410261225e-06
+iteration 628 res= 5.710975980999371e-06
+iteration 629 res= 5.636838208737748e-06
+iteration 630 res= 5.563662866019692e-06
+iteration 631 res= 5.491437458910295e-06
+iteration 632 res= 5.420149655664635e-06
+iteration 633 res= 5.349787284654071e-06
+iteration 634 res= 5.280338332221392e-06
+iteration 635 res= 5.211790940701037e-06
+iteration 636 res= 5.144133406330311e-06
+iteration 637 res= 5.077354177312871e-06
+iteration 638 res= 5.011441851789589e-06
+iteration 639 res= 4.946385175922334e-06
+iteration 640 res= 4.882173041978298e-06
+iteration 641 res= 4.818794486410211e-06
+iteration 642 res= 4.756238687980423e-06
+iteration 643 res= 4.6944949659603845e-06
+iteration 644 res= 4.633552778265063e-06
+iteration 645 res= 4.573401719666589e-06
+iteration 646 res= 4.5140315199942574e-06
+iteration 647 res= 4.455432042414259e-06
+iteration 648 res= 4.397593281683763e-06
+iteration 649 res= 4.340505362462486e-06
+iteration 650 res= 4.284158537600138e-06
+iteration 651 res= 4.228543186444524e-06
+iteration 652 res= 4.173649813297862e-06
+iteration 653 res= 4.119469045682479e-06
+iteration 654 res= 4.065991632806302e-06
+iteration 655 res= 4.013208443979321e-06
+iteration 656 res= 3.961110467043085e-06
+iteration 657 res= 3.909688806795636e-06
+iteration 658 res= 3.858934683558147e-06
+iteration 659 res= 3.808839431596989e-06
+iteration 660 res= 3.759394497689136e-06
+iteration 661 res= 3.710591439648555e-06
+iteration 662 res= 3.662421924847349e-06
+iteration 663 res= 3.6148777288780695e-06
+iteration 664 res= 3.567950734082638e-06
+iteration 665 res= 3.5216329281557663e-06
+iteration 666 res= 3.475916402849004e-06
+iteration 667 res= 3.4307933525349596e-06
+iteration 668 res= 3.3862560729624886e-06
+iteration 669 res= 3.3422969598601122e-06
+iteration 670 res= 3.2989085076650872e-06
+iteration 671 res= 3.25608330827523e-06
+iteration 672 res= 3.213814049752417e-06
+iteration 673 res= 3.1720935150636443e-06
+iteration 674 res= 3.130914580899014e-06
+iteration 675 res= 3.090270216362519e-06
+iteration 676 res= 3.050153481901479e-06
+iteration 677 res= 3.010557527991454e-06
+iteration 678 res= 2.9714755940607403e-06
+iteration 679 res= 2.9329010072798083e-06
+iteration 680 res= 2.894827181467167e-06
+iteration 681 res= 2.857247615928779e-06
+iteration 682 res= 2.8201558943373012e-06
+iteration 683 res= 2.78354568369719e-06
+iteration 684 res= 2.7474107332053174e-06
+iteration 685 res= 2.7117448732162838e-06
+iteration 686 res= 2.676542014161403e-06
+iteration 687 res= 2.6417961455121468e-06
+iteration 688 res= 2.6075013348413405e-06
+iteration 689 res= 2.573651726625357e-06
+iteration 690 res= 2.540241541449354e-06
+iteration 691 res= 2.5072650748571836e-06
+iteration 692 res= 2.4747166964797957e-06
+iteration 693 res= 2.4425908490411983e-06
+iteration 694 res= 2.4108820474059794e-06
+iteration 695 res= 2.3795848776214843e-06
+iteration 696 res= 2.3486939960356257e-06
+iteration 697 res= 2.3182041283845126e-06
+iteration 698 res= 2.2881100688307007e-06
+iteration 699 res= 2.2584066791552116e-06
+iteration 700 res= 2.2290888878152747e-06
+iteration 701 res= 2.2001516891196753e-06
+iteration 702 res= 2.1715901423608023e-06
+iteration 703 res= 2.1433993709578165e-06
+iteration 704 res= 2.115574561638924e-06
+iteration 705 res= 2.088110963635681e-06
+iteration 706 res= 2.0610038878150664e-06
+iteration 707 res= 2.03424870595914e-06
+iteration 708 res= 2.0078408498839025e-06
+iteration 709 res= 1.9817758107578653e-06
+iteration 710 res= 1.956049138237354e-06
+iteration 711 res= 1.930656439799365e-06
+iteration 712 res= 1.9055933798938076e-06
+iteration 713 res= 1.8808556792796004e-06
+iteration 714 res= 1.8564391142661972e-06
+iteration 715 res= 1.8323395160046549e-06
+iteration 716 res= 1.808552769731488e-06
+iteration 717 res= 1.7850748141097608e-06
+iteration 718 res= 1.7619016405572814e-06
+iteration 719 res= 1.7390292924914046e-06
+iteration 720 res= 1.7164538647058043e-06
+iteration 721 res= 1.6941715027077667e-06
+iteration 722 res= 1.6721784020101577e-06
+iteration 723 res= 1.6504708075341093e-06
+iteration 724 res= 1.6290450129445178e-06
+iteration 725 res= 1.6078973600313189e-06
+iteration 726 res= 1.5870242380343326e-06
+iteration 727 res= 1.5664220831276537e-06
+iteration 728 res= 1.546087377684961e-06
+iteration 729 res= 1.5260166498155485e-06
+iteration 730 res= 1.506206472627364e-06
+iteration 731 res= 1.4866534637537521e-06
+iteration 732 res= 1.4673542847468377e-06
+iteration 733 res= 1.4483056404685698e-06
+iteration 734 res= 1.4295042785731786e-06
+iteration 735 res= 1.4109469889378439e-06
+iteration 736 res= 1.3926306031066111e-06
+iteration 737 res= 1.374551993741593e-06
+iteration 738 res= 1.3567080741273448e-06
+iteration 739 res= 1.3390957976165863e-06
+iteration 740 res= 1.32171215710075e-06
+iteration 741 res= 1.304554184508048e-06
+iteration 742 res= 1.2876189502958448e-06
+iteration 743 res= 1.2709035629679069e-06
+iteration 744 res= 1.2544051685532662e-06
+iteration 745 res= 1.2381209501211105e-06
+iteration 746 res= 1.2220481273287446e-06
+iteration 747 res= 1.2061839559180712e-06
+iteration 748 res= 1.1905257272426086e-06
+iteration 749 res= 1.175070767839945e-06
+iteration 750 res= 1.159816438948274e-06
+iteration 751 res= 1.1447601360349643e-06
+iteration 752 res= 1.1298992884295236e-06
+iteration 753 res= 1.1152313587913439e-06
+iteration 754 res= 1.100753842731722e-06
+iteration 755 res= 1.0864642683675184e-06
+iteration 756 res= 1.072360195919254e-06
+iteration 757 res= 1.0584392172590654e-06
+iteration 758 res= 1.0446989555401453e-06
+iteration 759 res= 1.031137064758521e-06
+iteration 760 res= 1.0177512293694703e-06
+iteration 761 res= 1.0045391638886739e-06
+iteration 762 res= 9.914986125040712e-07
+iteration 763 res= 9.786273486798398e-07
+iteration 764 res= 9.65923174789745e-07
+iteration 765 res= 9.533839217288325e-07
+iteration 766 res= 9.41007448566981e-07
+iteration 767 res= 9.287916421419137e-07
+iteration 768 res= 9.167344167576691e-07
+iteration 769 res= 9.048337137445249e-07
+iteration 770 res= 8.930875012135601e-07
+iteration 771 res= 8.814937736217121e-07
+iteration 772 res= 8.70050551465781e-07
+iteration 773 res= 8.587558809387126e-07
+iteration 774 res= 8.476078335939651e-07
+iteration 775 res= 8.366045060413001e-07
+iteration 776 res= 8.257440195539977e-07
+iteration 777 res= 8.150245198388459e-07
+iteration 778 res= 8.044441766604903e-07
+iteration 779 res= 7.940011835455355e-07
+iteration 780 res= 7.836937574551647e-07
+iteration 781 res= 7.735201385187113e-07
+iteration 782 res= 7.63478589687715e-07
+iteration 783 res= 7.535673964882327e-07
+iteration 784 res= 7.437848666908485e-07
+iteration 785 res= 7.341293300454614e-07
+iteration 786 res= 7.245991379510238e-07
+iteration 787 res= 7.151926632430337e-07
+iteration 788 res= 7.059082998647418e-07
+iteration 789 res= 6.967444626252041e-07
+iteration 790 res= 6.876995868820185e-07
+iteration 791 res= 6.787721283365234e-07
+iteration 792 res= 6.699605627361711e-07
+iteration 793 res= 6.61263385559801e-07
+iteration 794 res= 6.526791118803383e-07
+iteration 795 res= 6.442062760418592e-07
+iteration 796 res= 6.358434313782327e-07
+iteration 797 res= 6.275891500326755e-07
+iteration 798 res= 6.194420226910784e-07
+iteration 799 res= 6.114006583015173e-07
+iteration 800 res= 6.034636838799841e-07
+iteration 801 res= 5.956297443078457e-07
+iteration 802 res= 5.878975019984479e-07
+iteration 803 res= 5.80265636761217e-07
+iteration 804 res= 5.727328455520035e-07
+iteration 805 res= 5.652978422127835e-07
+iteration 806 res= 5.579593573024822e-07
+iteration 807 res= 5.507161378407451e-07
+iteration 808 res= 5.435669471774986e-07
+iteration 809 res= 5.365105646064486e-07
+iteration 810 res= 5.295457853495919e-07
+iteration 811 res= 5.226714202627502e-07
+iteration 812 res= 5.158862955982548e-07
+iteration 813 res= 5.091892529016409e-07
+iteration 814 res= 5.025791486881916e-07
+iteration 815 res= 4.960548543800898e-07
+iteration 816 res= 4.896152560153696e-07
+iteration 817 res= 4.832592541210727e-07
+iteration 818 res= 4.769857634398105e-07
+iteration 819 res= 4.7079371287922217e-07
+iteration 820 res= 4.646820451882889e-07
+iteration 821 res= 4.5864971688939097e-07
+iteration 822 res= 4.526956980199584e-07
+iteration 823 res= 4.4681897199810074e-07
+iteration 824 res= 4.410185354177404e-07
+iteration 825 res= 4.3529339794411104e-07
+iteration 826 res= 4.2964258205894674e-07
+iteration 827 res= 4.2406512295048685e-07
+iteration 828 res= 4.1856006833466033e-07
+iteration 829 res= 4.131264782703839e-07
+iteration 830 res= 4.077634250335324e-07
+iteration 831 res= 4.0246999294821156e-07
+iteration 832 res= 3.972452782210609e-07
+iteration 833 res= 3.920883887762294e-07
+iteration 834 res= 3.869984441457911e-07
+iteration 835 res= 3.819745752677198e-07
+iteration 836 res= 3.770159243792428e-07
+iteration 837 res= 3.721216448305431e-07
+iteration 838 res= 3.672909009990807e-07
+iteration 839 res= 3.625228680656974e-07
+iteration 840 res= 3.57816731959784e-07
+iteration 841 res= 3.531716891380694e-07
+iteration 842 res= 3.485869465435747e-07
+iteration 843 res= 3.440617213336741e-07
+iteration 844 res= 3.3959524090937027e-07
+iteration 845 res= 3.351867426673466e-07
+iteration 846 res= 3.308354739054158e-07
+iteration 847 res= 3.2654069168459826e-07
+iteration 848 res= 3.2230166270121206e-07
+iteration 849 res= 3.1811766322224894e-07
+iteration 850 res= 3.139879788464132e-07
+iteration 851 res= 3.0991190449898496e-07
+iteration 852 res= 3.0588874420818237e-07
+iteration 853 res= 3.019178110914827e-07
+iteration 854 res= 2.979984271375401e-07
+iteration 855 res= 2.9412992315126017e-07
+iteration 856 res= 2.9031163865018426e-07
+iteration 857 res= 2.865429216696684e-07
+iteration 858 res= 2.8282312877191166e-07
+iteration 859 res= 2.791516248334903e-07
+iteration 860 res= 2.755277829739581e-07
+iteration 861 res= 2.7195098448607363e-07
+iteration 862 res= 2.6842061865364287e-07
+iteration 863 res= 2.6493608270918825e-07
+iteration 864 res= 2.614967816956943e-07
+iteration 865 res= 2.5810212841277594e-07
+iteration 866 res= 2.5475154323110364e-07
+iteration 867 res= 2.514444541090153e-07
+iteration 868 res= 2.481802963712982e-07
+iteration 869 res= 2.449585127223426e-07
+iteration 870 res= 2.417785530421235e-07
+iteration 871 res= 2.3863987441382233e-07
+iteration 872 res= 2.35541940935692e-07
+iteration 873 res= 2.3248422366738694e-07
+iteration 874 res= 2.294662005432404e-07
+iteration 875 res= 2.264873562679284e-07
+iteration 876 res= 2.2354718222590015e-07
+iteration 877 res= 2.2064517642604858e-07
+iteration 878 res= 2.177808433745675e-07
+iteration 879 res= 2.1495369403636806e-07
+iteration 880 res= 2.1216324567066315e-07
+iteration 881 res= 2.0940902186920773e-07
+iteration 882 res= 2.0669055238071772e-07
+iteration 883 res= 2.040073730358525e-07
+iteration 884 res= 2.0135902572448505e-07
+iteration 885 res= 1.9874505826086886e-07
+iteration 886 res= 1.9616502435090782e-07
+iteration 887 res= 1.9361848346648242e-07
+iteration 888 res= 1.9110500082414582e-07
+iteration 889 res= 1.886241472796307e-07
+iteration 890 res= 1.8617549924401761e-07
+iteration 891 res= 1.8375863863023085e-07
+iteration 892 res= 1.8137315280346556e-07
+iteration 893 res= 1.7901863446379571e-07
+iteration 894 res= 1.7669468159207327e-07
+iteration 895 res= 1.7440089741102745e-07
+iteration 896 res= 1.7213689027292495e-07
+iteration 897 res= 1.6990227363159035e-07
+iteration 898 res= 1.6769666594312047e-07
+iteration 899 res= 1.6551969063017227e-07
+iteration 900 res= 1.6337097599046877e-07
+iteration 901 res= 1.612501551765293e-07
+iteration 902 res= 1.5915686604936692e-07
+iteration 903 res= 1.5709075122289694e-07
+iteration 904 res= 1.550514579266195e-07
+iteration 905 res= 1.530386379556695e-07
+iteration 906 res= 1.510519476794904e-07
+iteration 907 res= 1.490910478607792e-07
+iteration 908 res= 1.471556037134376e-07
+iteration 909 res= 1.4524528478287857e-07
+iteration 910 res= 1.433597648950403e-07
+iteration 911 res= 1.4149872212039868e-07
+iteration 912 res= 1.3966183869753014e-07
+iteration 913 res= 1.3784880101344037e-07
+iteration 914 res= 1.3605929950518057e-07
+iteration 915 res= 1.342930286234674e-07
+iteration 916 res= 1.325496868274182e-07
+iteration 917 res= 1.308289764506408e-07
+iteration 918 res= 1.2913060365984675e-07
+iteration 919 res= 1.2745427853281933e-07
+iteration 920 res= 1.257997148194768e-07
+iteration 921 res= 1.241666300253485e-07
+iteration 922 res= 1.225547453498616e-07
+iteration 923 res= 1.2096378553233383e-07
+iteration 924 res= 1.1939347898114852e-07
+iteration 925 res= 1.1784355755462442e-07
+iteration 926 res= 1.1631375662231583e-07
+iteration 927 res= 1.1480381499605677e-07
+iteration 928 res= 1.1331347486076992e-07
+iteration 929 res= 1.1184248177686723e-07
+iteration 930 res= 1.1039058455457627e-07
+iteration 931 res= 1.089575353352478e-07
+iteration 932 res= 1.0754308941771209e-07
+iteration 933 res= 1.0614700530384293e-07
+iteration 934 res= 1.0476904463323833e-07
+iteration 935 res= 1.0340897213841772e-07
+iteration 936 res= 1.0206655558858967e-07
+iteration 937 res= 1.0074156578830825e-07
+iteration 938 res= 9.943377652010205e-08
+iteration 939 res= 9.814296447048262e-08
+iteration 940 res= 9.686890926096437e-08
+iteration 941 res= 9.561139335802752e-08
+iteration 942 res= 9.437020205896127e-08
+iteration 943 res= 9.314512343816768e-08
+iteration 944 res= 9.193594833065106e-08
+iteration 945 res= 9.074247027633103e-08
+iteration 946 res= 8.956448551495209e-08
+iteration 947 res= 8.840179290879835e-08
+iteration 948 res= 8.725419393776228e-08
+iteration 949 res= 8.612149267084058e-08
+iteration 950 res= 8.500349570544356e-08
+iteration 951 res= 8.390001215988825e-08
+iteration 952 res= 8.281085362674035e-08
+iteration 953 res= 8.173583413738262e-08
+iteration 954 res= 8.067477015012882e-08
+iteration 955 res= 7.962748049464245e-08
+iteration 956 res= 7.859378636867517e-08
+iteration 957 res= 7.757351126868399e-08
+iteration 958 res= 7.656648099033074e-08
+iteration 959 res= 7.557252361284128e-08
+iteration 960 res= 7.459146941524576e-08
+iteration 961 res= 7.36231508936116e-08
+iteration 962 res= 7.266740272466076e-08
+iteration 963 res= 7.17240617125759e-08
+iteration 964 res= 7.079296681418705e-08
+iteration 965 res= 6.98739590271147e-08
+iteration 966 res= 6.896688145860318e-08
+iteration 967 res= 6.807157923622497e-08
+iteration 968 res= 6.71878994892726e-08
+iteration 969 res= 6.631569134349658e-08
+iteration 970 res= 6.545480587217022e-08
+iteration 971 res= 6.460509609496931e-08
+iteration 972 res= 6.376641693311768e-08
+iteration 973 res= 6.293862519527101e-08
+iteration 974 res= 6.212157953591003e-08
+iteration 975 res= 6.13151404496769e-08
+iteration 976 res= 6.051917027112104e-08
+iteration 977 res= 5.973353306320599e-08
+iteration 978 res= 5.895809470038425e-08
+iteration 979 res= 5.8192722806786e-08
+iteration 980 res= 5.743728667131854e-08
+iteration 981 res= 5.6691657319573185e-08
+iteration 982 res= 5.5955707460176004e-08
+iteration 983 res= 5.52293114188232e-08
+iteration 984 res= 5.451234518771491e-08
+iteration 985 res= 5.3804686338082167e-08
+iteration 986 res= 5.310621404655408e-08
+iteration 987 res= 5.24168090742716e-08
+iteration 988 res= 5.1736353688911296e-08
+iteration 989 res= 5.106473170722243e-08
+iteration 990 res= 5.0401828497501395e-08
+iteration 991 res= 4.9747530832483177e-08
+iteration 992 res= 4.9101727017908766e-08
+iteration 993 res= 4.846430679210046e-08
+iteration 994 res= 4.783516131670321e-08
+iteration 995 res= 4.7214183159824156e-08
+iteration 996 res= 4.6601266319444236e-08
+iteration 997 res= 4.599630612360671e-08
+iteration 998 res= 4.5399199313054744e-08
+iteration 999 res= 4.480984390289662e-08
+iteration 1000 res= 4.422813929642239e-08
+iteration 1001 res= 4.36539861560191e-08
+iteration 1002 res= 4.308728645129672e-08
+iteration 1003 res= 4.252794344367134e-08
+iteration 1004 res= 4.197586160223019e-08
+iteration 1005 res= 4.1430946699626475e-08
+iteration 1006 res= 4.0893105661516577e-08
+iteration 1007 res= 4.036224667253229e-08
+iteration 1008 res= 3.983827910799811e-08
+iteration 1009 res= 3.9321113487666565e-08
+iteration 1010 res= 3.881066152624395e-08
+iteration 1011 res= 3.8306836046527666e-08
+iteration 1012 res= 3.780955104794992e-08
+iteration 1013 res= 3.7318721613655894e-08
+iteration 1014 res= 3.683426394981024e-08
+iteration 1015 res= 3.635609533287405e-08
+iteration 1016 res= 3.588413412031272e-08
+iteration 1017 res= 3.541829972105971e-08
+iteration 1018 res= 3.495851261805644e-08
+iteration 1019 res= 3.4504694311200466e-08
+iteration 1020 res= 3.405676727939478e-08
+iteration 1021 res= 3.3614655096669315e-08
+iteration 1022 res= 3.317828222147949e-08
+iteration 1023 res= 3.2747574200991965e-08
+iteration 1024 res= 3.232245745217153e-08
+iteration 1025 res= 3.1902859409027845e-08
+iteration 1026 res= 3.148870843406171e-08
+iteration 1027 res= 3.107993381908141e-08
+iteration 1028 res= 3.0676465749153223e-08
+iteration 1029 res= 3.027823536927532e-08
+iteration 1030 res= 2.988517465743019e-08
+iteration 1031 res= 2.94972165122926e-08
+iteration 1032 res= 2.911429469254592e-08
+iteration 1033 res= 2.8736343819780777e-08
+iteration 1034 res= 2.8363299374637706e-08
+iteration 1035 res= 2.7995097647933755e-08
+iteration 1036 res= 2.7631675767592302e-08
+iteration 1037 res= 2.727297169951555e-08
+iteration 1038 res= 2.6918924196569272e-08
+iteration 1039 res= 2.6569472811164417e-08
+iteration 1040 res= 2.6224557858912263e-08
+iteration 1041 res= 2.588412046706853e-08
+iteration 1042 res= 2.5548102509896967e-08
+iteration 1043 res= 2.521644661006882e-08
+iteration 1044 res= 2.4889096157216683e-08
+iteration 1045 res= 2.456599522381861e-08
+iteration 1046 res= 2.4247088680089034e-08
+iteration 1047 res= 2.393232205380205e-08
+iteration 1048 res= 2.362164161618994e-08
+iteration 1049 res= 2.3314994304409677e-08
+iteration 1050 res= 2.301232778097442e-08
+iteration 1051 res= 2.2713590362096582e-08
+iteration 1052 res= 2.241873103843976e-08
+iteration 1053 res= 2.2127699469435656e-08
+iteration 1054 res= 2.1840445961323125e-08
+iteration 1055 res= 2.1556921468896158e-08
+iteration 1056 res= 2.1277077594782447e-08
+iteration 1057 res= 2.100086653382423e-08
+iteration 1058 res= 2.0728241151096923e-08
+iteration 1059 res= 2.0459154888953055e-08
+iteration 1060 res= 2.0193561810259215e-08
+iteration 1061 res= 1.9931416550498665e-08
+iteration 1062 res= 1.9672674359172763e-08
+iteration 1063 res= 1.9417291066199173e-08
+iteration 1064 res= 1.9165223068871754e-08
+iteration 1065 res= 1.8916427324824442e-08
+iteration 1066 res= 1.86708613481524e-08
+iteration 1067 res= 1.842848321740496e-08
+iteration 1068 res= 1.818925155183267e-08
+iteration 1069 res= 1.7953125500651935e-08
+iteration 1070 res= 1.772006474503928e-08
+iteration 1071 res= 1.7490029516503925e-08
+iteration 1072 res= 1.7262980493165072e-08
+iteration 1073 res= 1.7038878943223295e-08
+iteration 1074 res= 1.681768660522311e-08
+iteration 1075 res= 1.659936568544082e-08
+iteration 1076 res= 1.6383878944716526e-08
+iteration 1077 res= 1.6171189563314722e-08
+iteration 1078 res= 1.5961261230131722e-08
+iteration 1079 res= 1.575405811632668e-08
+iteration 1080 res= 1.5549544837804438e-08
+iteration 1081 res= 1.5347686468577636e-08
+iteration 1082 res= 1.5148448551236176e-08
+iteration 1083 res= 1.4951797069549447e-08
+iteration 1084 res= 1.4757698441464384e-08
+iteration 1085 res= 1.456611951991461e-08
+iteration 1086 res= 1.4377027616988962e-08
+iteration 1087 res= 1.4190390420495433e-08
+iteration 1088 res= 1.4006176087112749e-08
+iteration 1089 res= 1.3824353162018285e-08
+iteration 1090 res= 1.3644890583558218e-08
+iteration 1091 res= 1.3467757729772602e-08
+iteration 1092 res= 1.329292434614809e-08
+iteration 1093 res= 1.3120360574717125e-08
+iteration 1094 res= 1.2950036966547906e-08
+iteration 1095 res= 1.2781924428747101e-08
+iteration 1096 res= 1.2615994274907387e-08
+iteration 1097 res= 1.245221816473072e-08
+iteration 1098 res= 1.2290568130794858e-08
+iteration 1099 res= 1.2131016581142873e-08
+iteration 1100 res= 1.1973536271251303e-08
+iteration 1101 res= 1.1818100308622932e-08
+iteration 1102 res= 1.166468214734026e-08
+iteration 1103 res= 1.1513255624233392e-08
+iteration 1104 res= 1.1363794843504644e-08
+iteration 1105 res= 1.1216274307419146e-08
+iteration 1106 res= 1.1070668843470895e-08
+iteration 1107 res= 1.0926953570887493e-08
+iteration 1108 res= 1.0785103944648804e-08
+iteration 1109 res= 1.0645095764984172e-08
+iteration 1110 res= 1.050690512131717e-08
+iteration 1111 res= 1.0370508413637978e-08
+iteration 1112 res= 1.023588235798268e-08
+iteration 1113 res= 1.0103003961414168e-08
+iteration 1114 res= 9.971850555755032e-09
+iteration 1115 res= 9.842399718353708e-09
+iteration 1116 res= 9.714629377896689e-09
+iteration 1117 res= 9.588517694568607e-09
+iteration 1118 res= 9.46404315996261e-09
+iteration 1119 res= 9.341184483992735e-09
+iteration 1120 res= 9.219920737761604e-09
+iteration 1121 res= 9.100231175676368e-09
+iteration 1122 res= 8.98209538011545e-09
+iteration 1123 res= 8.865493183950311e-09
+iteration 1124 res= 8.750404661664295e-09
+iteration 1125 res= 8.636810196938522e-09
+iteration 1126 res= 8.52469036439943e-09
+iteration 1127 res= 8.414026013635029e-09
+iteration 1128 res= 8.304798278125529e-09
+iteration 1129 res= 8.196988504984766e-09
+iteration 1130 res= 8.090578273654667e-09
+iteration 1131 res= 7.985549400623082e-09
+iteration 1132 res= 7.881884003011943e-09
+iteration 1133 res= 7.779564330629435e-09
+iteration 1134 res= 7.678572945151581e-09
+iteration 1135 res= 7.578892582878857e-09
+iteration 1136 res= 7.480506238050972e-09
+iteration 1137 res= 7.383397113623191e-09
+iteration 1138 res= 7.287548611038169e-09
+iteration 1139 res= 7.192944387906884e-09
+iteration 1140 res= 7.099568276521837e-09
+iteration 1141 res= 7.00740433703757e-09
+iteration 1142 res= 6.916436848010067e-09
+iteration 1143 res= 6.8266502499548854e-09
+iteration 1144 res= 6.738029236061521e-09
+iteration 1145 res= 6.650558660353958e-09
+iteration 1146 res= 6.564223592195401e-09
+iteration 1147 res= 6.479009307730671e-09
+iteration 1148 res= 6.394901229732741e-09
+iteration 1149 res= 6.31188501755806e-09
+iteration 1150 res= 6.229946492889301e-09
+iteration 1151 res= 6.149071654216649e-09
+iteration 1152 res= 6.069246718299237e-09
+iteration 1153 res= 5.990458021283524e-09
+iteration 1154 res= 5.912692148281219e-09
+iteration 1155 res= 5.835935793004998e-09
+iteration 1156 res= 5.760175855588762e-09
+iteration 1157 res= 5.68539941049925e-09
+iteration 1158 res= 5.611593682835221e-09
+iteration 1159 res= 5.5387460797165794e-09
+iteration 1160 res= 5.4668441533509585e-09
+iteration 1161 res= 5.395875628191834e-09
+iteration 1162 res= 5.325828390794737e-09
+iteration 1163 res= 5.256690484935508e-09
+iteration 1164 res= 5.188450096677202e-09
+iteration 1165 res= 5.121095576990425e-09
+iteration 1166 res= 5.054615440033339e-09
+iteration 1167 res= 4.988998308658378e-09
+iteration 1168 res= 4.9242329995965085e-09
+iteration 1169 res= 4.860308457746531e-09
+iteration 1170 res= 4.7972137452938115e-09
+iteration 1171 res= 4.734938113981372e-09
+iteration 1172 res= 4.67347091976698e-09
+iteration 1173 res= 4.61280166172377e-09
+iteration 1174 res= 4.552920014429162e-09
+iteration 1175 res= 4.493815704749301e-09
+iteration 1176 res= 4.435478674314436e-09
+iteration 1177 res= 4.377898953167678e-09
+iteration 1178 res= 4.321066704150085e-09
+iteration 1179 res= 4.264972240042305e-09
+iteration 1180 res= 4.209605968337617e-09
+iteration 1181 res= 4.154958436739667e-09
+iteration 1182 res= 4.101020317973881e-09
+iteration 1183 res= 4.047782414885064e-09
+iteration 1184 res= 3.995235619453652e-09
+iteration 1185 res= 3.943370966931739e-09
+iteration 1186 res= 3.892179595042006e-09
+iteration 1187 res= 3.841652773394536e-09
+iteration 1188 res= 3.7917818795259225e-09
+iteration 1189 res= 3.7425583934710095e-09
+iteration 1190 res= 3.693973908032886e-09
+iteration 1191 res= 3.646020116628433e-09
+iteration 1192 res= 3.5986888457696455e-09
+iteration 1193 res= 3.551972025315765e-09
+iteration 1194 res= 3.505861645811342e-09
+iteration 1195 res= 3.4603498626668065e-09
+iteration 1196 res= 3.4154288947641627e-09
+iteration 1197 res= 3.3710910916703852e-09
+iteration 1198 res= 3.327328847440339e-09
+iteration 1199 res= 3.2841347056193933e-09
+iteration 1200 res= 3.2415012968116816e-09
+iteration 1201 res= 3.1994213399812857e-09
+iteration 1202 res= 3.1578876439133675e-09
+iteration 1203 res= 3.1168931372188793e-09
+iteration 1204 res= 3.0764307956698982e-09
+iteration 1205 res= 3.0364937153231383e-09
+iteration 1206 res= 2.9970750948707538e-09
+iteration 1207 res= 2.958168186242774e-09
+iteration 1208 res= 2.9197663600145824e-09
+iteration 1209 res= 2.8818630475428715e-09
+iteration 1210 res= 2.8444517733597933e-09
+iteration 1211 res= 2.8075261809476306e-09
+iteration 1212 res= 2.7710799209787263e-09
+iteration 1213 res= 2.7351067958198207e-09
+iteration 1214 res= 2.6996006606095164e-09
+iteration 1215 res= 2.6645554496865154e-09
+iteration 1216 res= 2.6299651834719593e-09
+iteration 1217 res= 2.595823974757301e-09
+iteration 1218 res= 2.5621259629917414e-09
+iteration 1219 res= 2.5288653989630265e-09
+iteration 1220 res= 2.496036611910615e-09
+iteration 1221 res= 2.463634005788076e-09
+iteration 1222 res= 2.4316520239539873e-09
+iteration 1223 res= 2.4000852309030366e-09
+iteration 1224 res= 2.3689282300009717e-09
+iteration 1225 res= 2.338175691938477e-09
+iteration 1226 res= 2.3078223562491753e-09
+iteration 1227 res= 2.2778630783000486e-09
+iteration 1228 res= 2.2482927182689054e-09
+iteration 1229 res= 2.2191062309765444e-09
+iteration 1230 res= 2.190298610574434e-09
+iteration 1231 res= 2.1618649775621814e-09
+iteration 1232 res= 2.133800463084036e-09
+iteration 1233 res= 2.1061002655211708e-09
+iteration 1234 res= 2.078759653296463e-09
+iteration 1235 res= 2.0517739822632677e-09
+iteration 1236 res= 2.0251386172716043e-09
+iteration 1237 res= 1.9988490225161136e-09
+iteration 1238 res= 1.9729007261675322e-09
+iteration 1239 res= 1.9472892521651783e-09
+iteration 1240 res= 1.9220102816226745e-09
+iteration 1241 res= 1.89705945282106e-09
+iteration 1242 res= 1.872432540752019e-09
+iteration 1243 res= 1.848125325840977e-09
+iteration 1244 res= 1.8241336537330319e-09
+iteration 1245 res= 1.800453434432252e-09
+iteration 1246 res= 1.777080620684552e-09
+iteration 1247 res= 1.7540112306253991e-09
+iteration 1248 res= 1.7312413088291043e-09
+iteration 1249 res= 1.7087669880486432e-09
+iteration 1250 res= 1.686584409516392e-09
+iteration 1251 res= 1.6646897980858589e-09
+iteration 1252 res= 1.6430794211078792e-09
+iteration 1253 res= 1.6217495799595046e-09
+iteration 1254 res= 1.6006966468141573e-09
+iteration 1255 res= 1.5799169930050997e-09
+iteration 1256 res= 1.5594071021979355e-09
+iteration 1257 res= 1.539163455013054e-09
+iteration 1258 res= 1.519182616182486e-09
+iteration 1259 res= 1.4994611651214765e-09
+iteration 1260 res= 1.4799957143176704e-09
+iteration 1261 res= 1.4607829647493513e-09
+iteration 1262 res= 1.4418196371618233e-09
+iteration 1263 res= 1.4231024642042687e-09
+iteration 1264 res= 1.4046282831683411e-09
+iteration 1265 res= 1.38639393019598e-09
+iteration 1266 res= 1.368396272018939e-09
+iteration 1267 res= 1.3506322747581438e-09
+iteration 1268 res= 1.333098871514263e-09
+iteration 1269 res= 1.3157930656667719e-09
+iteration 1270 res= 1.2987119417382352e-09
+iteration 1271 res= 1.2818525527296725e-09
+iteration 1272 res= 1.2652120252304182e-09
+iteration 1273 res= 1.2487875142109598e-09
+iteration 1274 res= 1.2325762176455876e-09
+iteration 1275 res= 1.2165753830377197e-09
+iteration 1276 res= 1.2007822500353177e-09
+iteration 1277 res= 1.1851941503810257e-09
+iteration 1278 res= 1.1698084044110546e-09
+iteration 1279 res= 1.1546223864927385e-09
+iteration 1280 res= 1.1396335097644343e-09
+iteration 1281 res= 1.124839228510706e-09
+iteration 1282 res= 1.1102369758661528e-09
+iteration 1283 res= 1.0958242912805948e-09
+iteration 1284 res= 1.0815987160701734e-09
+iteration 1285 res= 1.0675578078415776e-09
+iteration 1286 res= 1.05369916762203e-09
+iteration 1287 res= 1.0400204439164152e-09
+iteration 1288 res= 1.026519279261919e-09
+iteration 1289 res= 1.013193398867737e-09
+iteration 1290 res= 1.0000404971065147e-09
+iteration 1291 res= 9.870583551189648e-10
+iteration 1292 res= 9.742447252363516e-10
+iteration 1293 res= 9.615974455019116e-10
+iteration 1294 res= 9.49114346968807e-10
+iteration 1295 res= 9.367932914486253e-10
+iteration 1296 res= 9.246321990731759e-10
+iteration 1297 res= 9.12628976658361e-10
+iteration 1298 res= 9.007815676247171e-10
+iteration 1299 res= 8.890879540100502e-10
+iteration 1300 res= 8.775461514272679e-10
+iteration 1301 res= 8.661541657689113e-10
+iteration 1302 res= 8.549100828226637e-10
+iteration 1303 res= 8.438119741728691e-10
+iteration 1304 res= 8.328579022509608e-10
+iteration 1305 res= 8.220460665867309e-10
+iteration 1306 res= 8.113745798925344e-10
+iteration 1307 res= 8.008416159149219e-10
+iteration 1308 res= 7.904453825418549e-10
+iteration 1309 res= 7.801841130761352e-10
+iteration 1310 res= 7.70056060088539e-10
+iteration 1311 res= 7.600594799042021e-10
+iteration 1312 res= 7.501926733066821e-10
+iteration 1313 res= 7.404539547096464e-10
+iteration 1314 res= 7.308416535141692e-10
+iteration 1315 res= 7.213541435353381e-10
+iteration 1316 res= 7.119897839099873e-10
+iteration 1317 res= 7.02747001317795e-10
+iteration 1318 res= 6.936241972700129e-10
+iteration 1319 res= 6.846198373963453e-10
+iteration 1320 res= 6.757323551582142e-10
+iteration 1321 res= 6.669602569909082e-10
+iteration 1322 res= 6.58302023707013e-10
+iteration 1323 res= 6.497562028949333e-10
+iteration 1324 res= 6.413212957362293e-10
+iteration 1325 res= 6.329959192245385e-10
+iteration 1326 res= 6.24778594087627e-10
+iteration 1327 res= 6.166679508952812e-10
+iteration 1328 res= 6.086626048022428e-10
+iteration 1329 res= 6.007611827467131e-10
+iteration 1330 res= 5.929623122980923e-10
+iteration 1331 res= 5.852647055648424e-10
+iteration 1332 res= 5.776670134397215e-10
+iteration 1333 res= 5.701679578108335e-10
+iteration 1334 res= 5.627662495184974e-10
+iteration 1335 res= 5.554606355762341e-10
+iteration 1336 res= 5.482498447151878e-10
+iteration 1337 res= 5.411326646350475e-10
+iteration 1338 res= 5.341078929751189e-10
+iteration 1339 res= 5.271743033605252e-10
+iteration 1340 res= 5.203307275031184e-10
+iteration 1341 res= 5.135759880401036e-10
+iteration 1342 res= 5.0690892949503e-10
+iteration 1343 res= 5.003284327280499e-10
+iteration 1344 res= 4.938333532519353e-10
+iteration 1345 res= 4.87422598638365e-10
+iteration 1346 res= 4.810950570331532e-10
+iteration 1347 res= 4.748496620797251e-10
+iteration 1348 res= 4.68685340863268e-10
+iteration 1349 res= 4.6260105019218426e-10
+iteration 1350 res= 4.5659573351660355e-10
+iteration 1351 res= 4.5066837382748276e-10
+iteration 1352 res= 4.448179677383837e-10
+iteration 1353 res= 4.3904350067385286e-10
+iteration 1354 res= 4.333440090497214e-10
+iteration 1355 res= 4.277184919459048e-10
+iteration 1356 res= 4.221660229021426e-10
+iteration 1357 res= 4.1668562367092893e-10
+iteration 1358 res= 4.112763568918318e-10
+iteration 1359 res= 4.059373222262791e-10
+iteration 1360 res= 4.0066759811375635e-10
+iteration 1361 res= 3.9546628040122194e-10
+iteration 1362 res= 3.903324818419027e-10
+iteration 1363 res= 3.852653378440005e-10
+iteration 1364 res= 3.802639708049988e-10
+iteration 1365 res= 3.7532752393833294e-10
+iteration 1366 res= 3.7045516604558276e-10
+iteration 1367 res= 3.6564604795319943e-10
+iteration 1368 res= 3.608993656579479e-10
+iteration 1369 res= 3.562143156117888e-10
+iteration 1370 res= 3.5159007069854707e-10
+iteration 1371 res= 3.4702585849574456e-10
+iteration 1372 res= 3.4252089418443815e-10
+iteration 1373 res= 3.38074425320811e-10
+iteration 1374 res= 3.3368566744299526e-10
+iteration 1375 res= 3.29353887480829e-10
+iteration 1376 res= 3.2507833538269496e-10
+iteration 1377 res= 3.2085828478022067e-10
+iteration 1378 res= 3.1669302350560563e-10
+iteration 1379 res= 3.125818384969333e-10
+iteration 1380 res= 3.0852401628385564e-10
+iteration 1381 res= 3.045188749851736e-10
+iteration 1382 res= 3.0056573121471316e-10
+iteration 1383 res= 2.966638928221955e-10
+iteration 1384 res= 2.9281271878764146e-10
+needed 1384 iterations
+
    +
    +

    Experimenting with the mesh-size \(h\), we observe experimentally that the number of iterations and thus the condition number grow proportional to \(h^{-2}\). This can be easily calculated for a finite difference discretization on a uniform mesh. We will prove it for finite elements later.

    @@ -522,6 +1971,11 @@

    Experiments with the Richardson iteration +
    BaseWebGuiScene
+
    +
    +

    @@ -555,6 +2009,24 @@

    Experiments with the Richardson iteration
    + + +
    +

    previous

    +

    19. Basic Iterative Methods

    +
    +     + +
    +

    next

    +

    23. The Gradient Method

    +
    + +
    @@ -571,9 +2043,9 @@

    Experiments with the Richardson iteration @@ -599,7 +2071,7 @@

    Experiments with the Richardson iteration - © Copyright 2023. + © Copyright 2024.

    diff --git a/iterative/conjugategradients.html b/iterative/conjugategradients.html index bbbdd6e6..29a27156 100644 --- a/iterative/conjugategradients.html +++ b/iterative/conjugategradients.html @@ -8,7 +8,7 @@ - Conjugate Gradients — Introduction to Scientific Computing + 27. Conjugate Gradients — Introduction to Scientific Computing @@ -32,7 +32,7 @@ - + @@ -59,9 +59,13 @@ + + + + @@ -123,8 +127,6 @@ - -
    @@ -174,7 +176,7 @@ @@ -184,6 +186,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -332,8 +366,8 @@

    Contents

    @@ -346,7 +380,7 @@

    Contents

    -

    Conjugate Gradients#

    +

    27. Conjugate Gradients#

    The gradient method is an improvement to the Richardson method since it does not need the spectral bounds explicitly. The Chebyshev method is an improvement to the Richardson method since the number of iterations are reduced from \((\log \varepsilon^{-1}) \, \kappa\) to \((\log \varepsilon^{-1}) \, \sqrt{\kappa}\).

    The conjugate gradients method combines both advantages.

    All four methods need one matrix-vector multiplication with the system matrix \(A\), and one preconditioner application \(C^{-1}\) per iteration, plus some vector updates. The gradient method and conjugate gradient method need inner products.

    @@ -363,7 +397,7 @@

    Conjugate Gradients\({\mathcal K}^n\).

    Since every polynomial iteration (such as the Chebyshev method) produces iterates \(x^n \in {\mathcal K}^n\), we obtain error estimates for the cg-method immediately from the Chebyshev method.

    -

    Solving the minimization problem#

    +

    27.1. Solving the minimization problem#

    Assume an \(n\)-dimensional sub-space \(V^n \subset {\mathbb R}^N\) is spanned by basis vectors \(p_1, \ldots, p_n\). Let \(P \in {\mathbb R}^{N \times n}\) contain the basis vectors \(p^k\) as column vectors. We want to solve the minimization problem

    \[ @@ -401,7 +435,7 @@

    Solving the minimization problem -

    Expanding the Krylov-space#

    +

    27.2. Expanding the Krylov-space#

    Assume we are given the Krylov space \({\mathcal K}^n\), and we have computed the minimizer \(x_n \in {\mathcal K}^n\).

    The (negative) gradient of \(f(x) = \tfrac{1}{2} x^T A x - b^T x\) is

    @@ -499,6 +533,55 @@

    Expanding the Krylov-space +
    Iteration 0 err= 0.04676402896143131
+Iteration 1 err= 0.017580352461756998
+Iteration 2 err= 0.010943892997105303
+Iteration 3 err= 0.010322892607099909
+Iteration 4 err= 0.010764872130030715
+Iteration 5 err= 0.009281795630372259
+Iteration 6 err= 0.007992279738830611
+Iteration 7 err= 0.005009074451734905
+Iteration 8 err= 0.0032697262056305197
+Iteration 9 err= 0.002075465517785042
+Iteration 10 err= 0.0016131225802585387
+Iteration 11 err= 0.0015200805270048218
+Iteration 12 err= 0.001144447282165303
+Iteration 13 err= 0.0006959910114937423
+Iteration 14 err= 0.00038612986237379615
+Iteration 15 err= 0.0002553212632138624
+Iteration 16 err= 0.00015512810242169684
+Iteration 17 err= 0.00010814417620694034
+Iteration 18 err= 8.13303781027281e-05
+Iteration 19 err= 3.826975323951041e-05
+Iteration 20 err= 2.2428817354335567e-05
+Iteration 21 err= 1.6390634682798343e-05
+Iteration 22 err= 1.1480834339309933e-05
+Iteration 23 err= 1.0673024370044995e-05
+Iteration 24 err= 9.731099074150096e-06
+Iteration 25 err= 7.736757922979457e-06
+Iteration 26 err= 4.068446816863081e-06
+Iteration 27 err= 2.4452296505594806e-06
+Iteration 28 err= 1.5080310510392442e-06
+Iteration 29 err= 1.0173737625111862e-06
+Iteration 30 err= 5.323042263653669e-07
+Iteration 31 err= 2.746942779033556e-07
+Iteration 32 err= 1.581982850676611e-07
+Iteration 33 err= 9.371977980118657e-08
+Iteration 34 err= 4.369762835571992e-08
+Iteration 35 err= 2.2417318891126386e-08
+Iteration 36 err= 1.0247544417781502e-08
+Iteration 37 err= 5.580579974891575e-09
+Iteration 38 err= 4.032573826196892e-09
+Iteration 39 err= 2.8070000791894448e-09
+Iteration 40 err= 1.492964647373061e-09
+Iteration 41 err= 7.588905986993379e-10
+Iteration 42 err= 3.80227983277497e-10
+Iteration 43 err= 2.02673048769177e-10
+Iteration 44 err= 1.06693200656839e-10
+
    +
    +

    @@ -507,6 +590,11 @@

    Expanding the Krylov-space +
    BaseWebGuiScene
+
    +
    +

    @@ -541,6 +629,15 @@

    Expanding the Krylov-space
    + + +
    +

    previous

    +

    26. The Chebyshev Method

    +
    +
    @@ -557,8 +654,8 @@

    Expanding the Krylov-space @@ -583,7 +680,7 @@

    Expanding the Krylov-space - © Copyright 2023. + © Copyright 2024.

    diff --git a/iterative/gradientmethod.html b/iterative/gradientmethod.html index c172b731..29fdec95 100644 --- a/iterative/gradientmethod.html +++ b/iterative/gradientmethod.html @@ -8,7 +8,7 @@ - The Gradient Method — Introduction to Scientific Computing + 23. The Gradient Method — Introduction to Scientific Computing @@ -32,7 +32,7 @@ - + @@ -62,6 +62,8 @@ + + @@ -123,8 +125,6 @@ - -
    @@ -174,7 +174,7 @@ @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -335,7 +367,7 @@

    The Gradient Method

    -

    The Gradient Method#

    +

    23. The Gradient Method#

    In this section we assume that \(A\) is SPD.

    We define the function \(f : {\mathbb R}^n \rightarrow {\mathbb R}\) as $\( @@ -440,6 +472,728 @@

    The Gradient Method +
    iteration 0 res= 0.02948894632867606
+iteration 1 res= 0.06791995975712485
+iteration 2 res= 0.03300125136092373
+iteration 3 res= 0.026814591140556135
+iteration 4 res= 0.023337804574069775
+iteration 5 res= 0.023286509288959374
+iteration 6 res= 0.021355108513277838
+iteration 7 res= 0.02165226219690209
+iteration 8 res= 0.01999473366620558
+iteration 9 res= 0.02032568130615626
+iteration 10 res= 0.01880740525681351
+iteration 11 res= 0.01913452755048257
+iteration 12 res= 0.01772199686334556
+iteration 13 res= 0.01803833511145182
+iteration 14 res= 0.016716585272848326
+iteration 15 res= 0.017020689706431807
+iteration 16 res= 0.015780287582732772
+iteration 17 res= 0.016071994850263654
+iteration 18 res= 0.014905840870141079
+iteration 19 res= 0.015185341791197467
+iteration 20 res= 0.014087593720555187
+iteration 21 res= 0.01435519136227381
+iteration 22 res= 0.013320810906922019
+iteration 23 res= 0.013576849768606167
+iteration 24 res= 0.012601377631937262
+iteration 25 res= 0.012846220102887518
+iteration 26 res= 0.011925649091628234
+iteration 27 res= 0.012159666120065441
+iteration 28 res= 0.011290361332920355
+iteration 29 res= 0.01151392773041459
+iteration 30 res= 0.010692571469754457
+iteration 31 res= 0.010906062899525079
+iteration 32 res= 0.010129613669028021
+iteration 33 res= 0.010333404364765975
+iteration 34 res= 0.009599064562401591
+iteration 35 res= 0.009793525457649286
+iteration 36 res= 0.009098714869233809
+iteration 37 res= 0.009284212023383023
+iteration 38 res= 0.008626545466392178
+iteration 39 res= 0.008803438739940465
+iteration 40 res= 0.008180706854560684
+iteration 41 res= 0.008349348806693196
+iteration 42 res= 0.007759501342716949
+iteration 43 res= 0.00792023632985912
+iteration 44 res= 0.0073613674783874605
+iteration 45 res= 0.007514530933685438
+iteration 46 res= 0.006984866372974708
+iteration 47 res= 0.007130784247150682
+iteration 48 res= 0.00662866964848084
+iteration 49 res= 0.006767657993164756
+iteration 50 res= 0.006291548784145773
+iteration 51 res= 0.006423913459772389
+iteration 52 res= 0.005972365679134899
+iteration 53 res= 0.006098402170747008
+iteration 54 res= 0.0056700642759645835
+iteration 55 res= 0.005790057601668
+iteration 56 res= 0.005383663111924455
+iteration 57 res= 0.005497887810201045
+iteration 58 res= 0.005112248684144848
+iteration 59 res= 0.005220968867677012
+iteration 60 res= 0.004854969529269907
+iteration 61 res= 0.00495843899431734
+iteration 62 res= 0.004611030931643515
+iteration 63 res= 0.004709493313313558
+iteration 64 res= 0.004379690184978928
+iteration 65 res= 0.004473379149930562
+iteration 66 res= 0.0041602523420074325
+iteration 67 res= 0.004249391811220281
+iteration 68 res= 0.003952066394841866
+iteration 69 res= 0.00403687079006556
+iteration 70 res= 0.0037545218359454234
+iteration 71 res= 0.0038351963443247184
+iteration 72 res= 0.0035670455558219952
+iteration 73 res= 0.003643786407975383
+iteration 74 res= 0.0033890990389708656
+iteration 75 res= 0.003462093796491849
+iteration 76 res= 0.003220175824383241
+iteration 77 res= 0.0032896036733418193
+iteration 78 res= 0.003059799200992042
+iteration 79 res= 0.0031258312485470357
+iteration 80 res= 0.0029075201120976157
+iteration 81 res= 0.0029703196837956836
+iteration 82 res= 0.0027629152459476155
+iteration 83 res= 0.0028226381816894995
+iteration 84 res= 0.0026255852924073697
+iteration 85 res= 0.002682380239412835
+iteration 86 res= 0.0024951533480683757
+iteration 87 res= 0.0025491620494744733
+iteration 88 res= 0.0023712634542508767
+iteration 89 res= 0.0024226210322394436
+iteration 90 res= 0.0022535792542005026
+iteration 91 res= 0.0023024144867751666
+iteration 92 res= 0.00214178275739208
+iteration 93 res= 0.0021882183481170842
+iteration 94 res= 0.002035573200265282
+iteration 95 res= 0.002079726040442311
+iteration 96 res= 0.0019346659939524686
+iteration 97 res= 0.0019766474168509307
+iteration 98 res= 0.001838791750641121
+iteration 99 res= 0.0018787077775152562
+iteration 100 res= 0.0017476953811613378
+iteration 101 res= 0.0017856469588869192
+iteration 102 res= 0.0016611352572199133
+iteration 103 res= 0.0016972184874666674
+iteration 104 res= 0.0015788824324314795
+iteration 105 res= 0.0016131887923568798
+iteration 106 res= 0.0015007199169370365
+iteration 107 res= 0.0015333364714446994
+iteration 108 res= 0.001426442000962263
+iteration 109 res= 0.0014574516066155289
+iteration 110 res= 0.0013558536231621354
+iteration 111 res= 0.0013853351238819226
+iteration 112 res= 0.0012887697800332365
+iteration 113 res= 0.0013167981947402657
+iteration 114 res= 0.001225014973058191
+iteration 115 res= 0.0012516616754442167
+iteration 116 res= 0.0011644226905844477
+iteration 117 res= 0.0011897555812162251
+iteration 118 res= 0.001106834921737884
+iteration 119 res= 0.0011309185927120076
+iteration 120 res= 0.001052101699935345
+iteration 121 res= 0.001074997592312579
+iteration 122 res= 0.0010000806737936298
+iteration 123 res= 0.0010218472280484991
+iteration 124 res= 0.0009506367034393103
+iteration 125 res= 0.0009713295031651033
+iteration 126 res= 0.0009036414804074749
+iteration 127 res= 0.0009233133895188196
+iteration 128 res= 0.0008589731694808065
+iteration 129 res= 0.0008776744631560949
+iteration 130 res= 0.000816516070965909
+iteration 131 res= 0.0008342945605703206
+iteration 132 res= 0.0007761603020335981
+iteration 133 res= 0.0007930614542606701
+iteration 134 res= 0.0007378014958659223
+iteration 135 res= 0.000753868546331724
+iteration 136 res= 0.0007013405174565911
+iteration 137 res= 0.0007166145789758281
+iteration 138 res= 0.0006666831950047311
+iteration 139 res= 0.0006812033607727018
+iteration 140 res= 0.0006337400659257233
+iteration 141 res= 0.0006475435078241309
+iteration 142 res= 0.0006024261365783906
+iteration 143 res= 0.0006155481988166942
+iteration 144 res= 0.0005726606548759695
+iteration 145 res= 0.0005851349431733752
+iteration 146 res= 0.0005443668950099787
+iteration 147 res= 0.0005562253615163847
+iteration 148 res= 0.0005174719535719682
+iteration 149 res= 0.0005287449777193041
+iteration 150 res= 0.0004919065564089368
+iteration 151 res= 0.0005026230218773851
+iteration 152 res= 0.00046760487559439613
+iteration 153 res= 0.00047779224357107395
+iteration 154 res= 0.0004445043559392307
+iteration 155 res= 0.00045418873484003155
+iteration 156 res= 0.00042254555050504067
+iteration 157 res= 0.00043175176232354567
+iteration 158 res= 0.0004016719646179347
+iteration 159 res= 0.00041042360805864145
+iteration 160 res= 0.0003818299079131465
+iteration 161 res= 0.00039014941845973063
+iteration 162 res= 0.00036296835397062047
+iteration 163 res= 0.00037087706103357563
+iteration 164 res= 0.0003450388071291648
+iteration 165 res= 0.0003525569884109709
+iteration 166 res= 0.0003279951760920909
+iteration 167 res= 0.0003351421093020406
+iteration 168 res= 0.0003117936539606835
+iteration 169 res= 0.00031858766600567694
+iteration 170 res= 0.0002963926043535398
+iteration 171 res= 0.0003028511181255234
+iteration 172 res= 0.00028175245328994434
+iteration 173 res= 0.00028789203216523883
+iteration 174 res= 0.0002678355865341537
+iteration 175 res= 0.00027367197669468006
+iteration 176 res= 0.00025460625211487417
+iteration 177 res= 0.00026015442279625153
+iteration 178 res= 0.00024203046775044315
+iteration 179 res= 0.0002473046495170916
+iteration 180 res= 0.000230075932925373
+iteration 181 res= 0.0002350896540681067
+iteration 182 res= 0.00021871194537806924
+iteration 183 res= 0.00022347806652520908
+iteration 184 res= 0.00020790932177278012
+iteration 185 res= 0.00021244006880154502
+iteration 186 res= 0.00019764032234124534
+iteration 187 res= 0.0002019473176720891
+iteration 188 res= 0.00018787857929114597
+iteration 189 res= 0.00019197287164379506
+iteration 190 res= 0.00017859902878938295
+iteration 191 res= 0.00018249112147558657
+iteration 192 res= 0.00016977784633847746
+iteration 193 res= 0.00017347772416290386
+iteration 194 res= 0.00016139238537403987
+iteration 195 res= 0.00016490954021133612
+iteration 196 res= 0.00015342111892034327
+iteration 197 res= 0.0001567645740331113
+iteration 198 res= 0.0001458435841495981
+iteration 199 res= 0.00014902191730892608
+iteration 200 res= 0.00013864032969859431
+iteration 201 res= 0.00014166169516580918
+iteration 202 res= 0.00013179286560399212
+iteration 203 res= 0.0001346650150294649
+iteration 204 res= 0.00012528361572472716
+iteration 205 res= 0.00012801391801685983
+iteration 206 res= 0.0001190958725267835
+iteration 207 res= 0.00012169133274172885
+iteration 208 res= 0.00011321375411200166
+iteration 209 res= 0.00011568103141221169
+iteration 210 res= 0.00010762216337865107
+iteration 211 res= 0.00010996758810601088
+iteration 212 res= 0.00010230674920723285
+iteration 213 res= 0.00010453633911430603
+iteration 214 res= 9.725386957040364e-05
+iteration 215 res= 9.937334525119363e-05
+iteration 216 res= 9.245055647104966e-05
+iteration 217 res= 9.446535603065962e-05
+iteration 218 res= 8.7884482617403e-05
+iteration 219 res= 8.979977561805117e-05
+iteration 220 res= 8.354392974870069e-05
+iteration 221 res= 8.536463046771659e-05
+iteration 222 res= 7.941775852925234e-05
+iteration 223 res= 8.114853856293415e-05
+iteration 224 res= 7.549537993291984e-05
+iteration 225 res= 7.714068017847356e-05
+iteration 226 res= 7.176672804393563e-05
+iteration 227 res= 7.33307700901369e-05
+iteration 228 res= 6.822223420370561e-05
+iteration 229 res= 6.970903115942069e-05
+iteration 230 res= 6.48528024367702e-05
+iteration 231 res= 6.626616922504345e-05
+iteration 232 res= 6.164978609244406e-05
+iteration 233 res= 6.299334923649793e-05
+iteration 234 res= 5.860496564183035e-05
+iteration 235 res= 5.98821725680282e-05
+iteration 236 res= 5.57105275729179e-05
+iteration 237 res= 5.6924655454510064e-05
+iteration 238 res= 5.2959044329330075e-05
+iteration 239 res= 5.411320849363053e-05
+iteration 240 res= 5.034345524100799e-05
+iteration 241 res= 5.1440617161535644e-05
+iteration 242 res= 4.785704839768829e-05
+iteration 243 res= 4.8900023291746633e-05
+iteration 244 res= 4.5493443418482053e-05
+iteration 245 res= 4.648490746964288e-05
+iteration 246 res= 4.3246575073184506e-05
+iteration 247 res= 4.418907229718326e-05
+iteration 248 res= 4.1110677713152156e-05
+iteration 249 res= 4.200662648479017e-05
+iteration 250 res= 3.908027047167835e-05
+iteration 251 res= 3.993196972946077e-05
+iteration 252 res= 3.7150143195789246e-05
+iteration 253 res= 3.7959778340202986e-05
+iteration 254 res= 3.5315343073271956e-05
+iteration 255 res= 3.6084991573823933e-05
+iteration 256 res= 3.357116192054294e-05
+iteration 257 res= 3.4302798645933496e-05
+iteration 258 res= 3.1913124098669753e-05
+iteration 259 res= 3.26086263837676e-05
+iteration 260 res= 3.03369750264812e-05
+iteration 261 res= 3.099812748909188e-05
+iteration 262 res= 2.8838670261239483e-05
+iteration 263 res= 2.946716938101907e-05
+iteration 264 res= 2.7414365118811876e-05
+iteration 265 res= 2.801182359006793e-05
+iteration 266 res= 2.606040480666925e-05
+iteration 267 res= 2.6628355676212224e-05
+iteration 268 res= 2.47733150443598e-05
+iteration 269 res= 2.531321564501679e-05
+iteration 270 res= 2.3549793147361368e-05
+iteration 271 res= 2.406302883724116e-05
+iteration 272 res= 2.2386699551408942e-05
+iteration 273 res= 2.2874587268508927e-05
+iteration 274 res= 2.128104975552709e-05
+iteration 275 res= 2.1744841396799558e-05
+iteration 276 res= 2.023000666307457e-05
+iteration 277 res= 2.067089229662e-05
+iteration 278 res= 1.9230873301132005e-05
+iteration 279 res= 1.964998421975903e-05
+iteration 280 res= 1.8281085899536646e-05
+iteration 281 res= 1.86794975235217e-05
+iteration 282 res= 1.7378207311792714e-05
+iteration 283 res= 1.7756941948285797e-05
+iteration 284 res= 1.6519920760964997e-05
+iteration 285 res= 1.6879950227120322e-05
+iteration 286 res= 1.57040238944984e-05
+iteration 287 res= 1.6046272011059373e-05
+iteration 288 res= 1.4928423132700251e-05
+iteration 289 res= 1.5253768094435814e-05
+iteration 290 res= 1.4191128296376766e-05
+iteration 291 res= 1.4500404925450517e-05
+iteration 292 res= 1.3490247499832086e-05
+iteration 293 res= 1.3784249387885247e-05
+iteration 294 res= 1.282398229612038e-05
+iteration 295 res= 1.3103463840564224e-05
+iteration 296 res= 1.2190623062089119e-05
+iteration 297 res= 1.2456301401831238e-05
+iteration 298 res= 1.1588544611367724e-05
+iteration 299 res= 1.1841101466938476e-05
+iteration 300 res= 1.1016202024041269e-05
+iteration 301 res= 1.1256285446841506e-05
+iteration 302 res= 1.0472126682305165e-05
+iteration 303 res= 1.0700352717463344e-05
+iteration 304 res= 9.954922501925904e-06
+iteration 305 res= 1.0171876769030931e-05
+iteration 306 res= 9.463262349835614e-06
+iteration 307 res= 9.669501545601191e-06
+iteration 308 res= 8.995884638666104e-06
+iteration 309 res= 9.19193796538207e-06
+iteration 310 res= 8.551590089482377e-06
+iteration 311 res= 8.73796061291806e-06
+iteration 312 res= 8.129238654407412e-06
+iteration 313 res= 8.30640459465114e-06
+iteration 314 res= 7.727746591240403e-06
+iteration 315 res= 7.896162549787221e-06
+iteration 316 res= 7.346083682561007e-06
+iteration 317 res= 7.5061818087970354e-06
+iteration 318 res= 6.9832705921828445e-06
+iteration 319 res= 7.135461692259338e-06
+iteration 320 res= 6.638376352172194e-06
+iteration 321 res= 6.783050943114457e-06
+iteration 322 res= 6.3105159739828845e-06
+iteration 323 res= 6.448045285738761e-06
+iteration 324 res= 5.998848177577032e-06
+iteration 325 res= 6.129585105576075e-06
+iteration 326 res= 5.7025732327040535e-06
+iteration 327 res= 5.826853243371508e-06
+iteration 328 res= 5.420930906798225e-06
+iteration 329 res= 5.539072898347295e-06
+iteration 330 res= 5.153198514228789e-06
+iteration 331 res= 5.265505634939879e-06
+iteration 332 res= 4.8986890618966795e-06
+iteration 333 res= 5.005449487983199e-06
+iteration 334 res= 4.656749486419162e-06
+iteration 335 res= 4.758237161475859e-06
+iteration 336 res= 4.426758978378869e-06
+iteration 337 res= 4.523234316310012e-06
+iteration 338 res= 4.2081273893369765e-06
+iteration 339 res= 4.2998379425680835e-06
+iteration 340 res= 4.00029371752284e-06
+iteration 341 res= 4.087474812210482e-06
+iteration 342 res= 3.802724668314205e-06
+iteration 343 res= 3.8856000081838e-06
+iteration 344 res= 3.614913285814063e-06
+iteration 345 res= 3.693695526175086e-06
+iteration 346 res= 3.436377652012726e-06
+iteration 347 res= 3.5112689454241666e-06
+iteration 348 res= 3.2666596501970495e-06
+iteration 349 res= 3.3378521651833164e-06
+iteration 350 res= 3.1053237894336867e-06
+iteration 351 res= 3.1730002035819116e-06
+iteration 352 res= 2.951956087109932e-06
+iteration 353 res= 3.016290055813929e-06
+iteration 354 res= 2.806163006664712e-06
+iteration 355 res= 2.8673196087183873e-06
+iteration 356 res= 2.667570447783909e-06
+iteration 357 res= 2.7257066089674622e-06
+iteration 358 res= 2.535822786468842e-06
+iteration 359 res= 2.591087682214674e-06
+iteration 360 res= 2.4105819625146926e-06
+iteration 361 res= 2.4631174006862345e-06
+iteration 362 res= 2.2915266120573344e-06
+iteration 363 res= 2.341467396822982e-06
+iteration 364 res= 2.178351242962676e-06
+iteration 365 res= 2.2258255206985165e-06
+iteration 366 res= 2.07076545094255e-06
+iteration 367 res= 2.115895039051459e-06
+iteration 368 res= 1.9684931743857065e-06
+iteration 369 res= 2.0113938738765385e-06
+iteration 370 res= 1.8712719859918164e-06
+iteration 371 res= 1.912053878620746e-06
+iteration 372 res= 1.7788524193907872e-06
+iteration 373 res= 1.8176201501272756e-06
+iteration 374 res= 1.6909973290195395e-06
+iteration 375 res= 1.727850374561701e-06
+iteration 376 res= 1.607481281613662e-06
+iteration 377 res= 1.642514205642038e-06
+iteration 378 res= 1.5280899777525392e-06
+iteration 379 res= 1.5613926735772451e-06
+iteration 380 res= 1.4526197019736257e-06
+iteration 381 res= 1.484277623197499e-06
+iteration 382 res= 1.3808768000448825e-06
+iteration 383 res= 1.4109711798345022e-06
+iteration 384 res= 1.3126771820540553e-06
+iteration 385 res= 1.3412852415812712e-06
+iteration 386 res= 1.2478458500397346e-06
+iteration 387 res= 1.2750409966285617e-06
+iteration 388 res= 1.1862164489521056e-06
+iteration 389 res= 1.2120684644394174e-06
+iteration 390 res= 1.1276308397911643e-06
+iteration 391 res= 1.1522060595845156e-06
+iteration 392 res= 1.071938693827074e-06
+iteration 393 res= 1.0953001771191073e-06
+iteration 394 res= 1.0189971068614522e-06
+iteration 395 res= 1.041204798437649e-06
+iteration 396 res= 9.686702325397808e-07
+iteration 397 res= 9.897811165947534e-07
+iteration 398 res= 9.20828933774035e-07
+iteration 399 res= 9.408971801310444e-07
+iteration 400 res= 8.753504513810833e-07
+iteration 401 res= 8.944275544899768e-07
+iteration 402 res= 8.321180890865988e-07
+iteration 403 res= 8.502530001568323e-07
+iteration 404 res= 7.910209140862082e-07
+iteration 405 res= 8.082601666940008e-07
+iteration 406 res= 7.519534723955252e-07
+iteration 407 res= 7.683413018874486e-07
+iteration 408 res= 7.148155182586634e-07
+iteration 409 res= 7.303939752580552e-07
+iteration 410 res= 6.795117569209044e-07
+iteration 411 res= 6.943208152283532e-07
+iteration 412 res= 6.459516001054739e-07
+iteration 413 res= 6.60029259270249e-07
+iteration 414 res= 6.140489335669948e-07
+iteration 415 res= 6.274313163926138e-07
+iteration 416 res= 5.837218961251595e-07
+iteration 417 res= 5.964433413592913e-07
+iteration 418 res= 5.548926696116326e-07
+iteration 419 res= 5.669858200581712e-07
+iteration 420 res= 5.274872791911928e-07
+iteration 421 res= 5.389831654705989e-07
+iteration 422 res= 5.01435403544743e-07
+iteration 423 res= 5.123635237175762e-07
+iteration 424 res= 4.76670194427125e-07
+iteration 425 res= 4.870585896850795e-07
+iteration 426 res= 4.5312810513673027e-07
+iteration 427 res= 4.6300343175539336e-07
+iteration 428 res= 4.3074872745676226e-07
+iteration 429 res= 4.401363251947264e-07
+iteration 430 res= 4.0947463664975147e-07
+iteration 431 res= 4.1839859376958594e-07
+iteration 432 res= 3.892512441075786e-07
+iteration 433 res= 3.9773445918550586e-07
+iteration 434 res= 3.7002665727891417e-07
+iteration 435 res= 3.780908979617911e-07
+iteration 436 res= 3.517515465146516e-07
+iteration 437 res= 3.5941750537502286e-07
+iteration 438 res= 3.3437901848966264e-07
+iteration 439 res= 3.4166636612221083e-07
+iteration 440 res= 3.178644958760812e-07
+iteration 441 res= 3.247919313717151e-07
+iteration 442 res= 3.0216560295935946e-07
+iteration 443 res= 3.0875090188645624e-07
+iteration 444 res= 2.8724205690359073e-07
+iteration 445 res= 2.9350211691951094e-07
+iteration 446 res= 2.73055564387091e-07
+iteration 447 res= 2.7900644859700284e-07
+iteration 448 res= 2.595697233430076e-07
+iteration 449 res= 2.6522670151727804e-07
+iteration 450 res= 2.467499295528256e-07
+iteration 451 res= 2.5212751730874146e-07
+iteration 452 res= 2.3456328785309372e-07
+iteration 453 res= 2.396752839014505e-07
+iteration 454 res= 2.229785277275286e-07
+iteration 455 res= 2.2783804927966214e-07
+iteration 456 res= 2.119659230679104e-07
+iteration 457 res= 2.1658543949402282e-07
+iteration 458 res= 2.0149721589787799e-07
+iteration 459 res= 2.0588858072302559e-07
+iteration 460 res= 1.9154554386390203e-07
+iteration 461 res= 1.9572002518374484e-07
+iteration 462 res= 1.8208537130738008e-07
+iteration 463 res= 1.8605368070173925e-07
+iteration 464 res= 1.7309242374098638e-07
+iteration 465 res= 1.768647437594005e-07
+iteration 466 res= 1.6454362556114597e-07
+iteration 467 res= 1.681296358509532e-07
+iteration 468 res= 1.564170408368041e-07
+iteration 469 res= 1.598259429807943e-07
+iteration 470 res= 1.4869181702255783e-07
+iteration 471 res= 1.5193235814992696e-07
+iteration 472 res= 1.4134813145171891e-07
+iteration 473 res= 1.4442862668291073e-07
+iteration 474 res= 1.3436714047201184e-07
+iteration 475 res= 1.3729549425503927e-07
+iteration 476 res= 1.277309310933895e-07
+iteration 477 res= 1.305146574863846e-07
+iteration 478 res= 1.214224750238976e-07
+iteration 479 res= 1.2406871697593416e-07
+iteration 480 res= 1.1542558497564515e-07
+iteration 481 res= 1.1794113265530753e-07
+iteration 482 res= 1.09724873128763e-07
+iteration 483 res= 1.1211618134749221e-07
+iteration 484 res= 1.0430571164677095e-07
+iteration 485 res= 1.0657891642169545e-07
+iteration 486 res= 9.91541951420369e-08
+iteration 487 res= 1.013151294407878e-07
+iteration 488 res= 9.425710499501555e-08
+iteration 489 res= 9.631131370292703e-08
+iteration 490 res= 8.960187543571136e-08
+iteration 491 res= 9.15546295838115e-08
+iteration 492 res= 8.517656130033166e-08
+iteration 493 res= 8.703287159063208e-08
+iteration 494 res= 8.096980738039443e-08
+iteration 495 res= 8.273443704318461e-08
+iteration 496 res= 7.697081928564206e-08
+iteration 497 res= 7.864829630177924e-08
+iteration 498 res= 7.316933574599566e-08
+iteration 499 res= 7.476396446555289e-08
+iteration 500 res= 6.95556022814783e-08
+iteration 501 res= 7.10714744685628e-08
+iteration 502 res= 6.612034617254447e-08
+iteration 503 res= 6.75613515046282e-08
+iteration 504 res= 6.285475266659099e-08
+iteration 505 res= 6.42245887152931e-08
+iteration 506 res= 5.975044235959575e-08
+iteration 507 res= 6.105262407852808e-08
+iteration 508 res= 5.679944969484631e-08
+iteration 509 res= 5.803731843886711e-08
+iteration 510 res= 5.399420252358717e-08
+iteration 511 res= 5.517093462260609e-08
+iteration 512 res= 5.1327502675138306e-08
+iteration 513 res= 5.244611758447313e-08
+iteration 514 res= 4.879250748662941e-08
+iteration 515 res= 4.9855875534827703e-08
+iteration 516 res= 4.638271224495513e-08
+iteration 517 res= 4.739356199896139e-08
+iteration 518 res= 4.4091933495898077e-08
+iteration 519 res= 4.505285876246566e-08
+iteration 520 res= 4.191429317759161e-08
+iteration 521 res= 4.2827759658904287e-08
+iteration 522 res= 3.9844203537609004e-08
+iteration 523 res= 4.071255515819049e-08
+iteration 524 res= 3.7876352794977175e-08
+iteration 525 res= 3.870181771612333e-08
+iteration 526 res= 3.600569151032373e-08
+iteration 527 res= 3.679038784749011e-08
+iteration 528 res= 3.422741962918403e-08
+iteration 529 res= 3.4973360886999476e-08
+iteration 530 res= 3.2536974165221506e-08
+iteration 531 res= 3.324607440407384e-08
+iteration 532 res= 3.093001749175682e-08
+iteration 533 res= 3.160409623920802e-08
+iteration 534 res= 2.940242621156253e-08
+iteration 535 res= 3.004321313119573e-08
+iteration 536 res= 2.795028057636331e-08
+iteration 537 res= 2.8559419906041864e-08
+iteration 538 res= 2.6569854428892624e-08
+iteration 539 res= 2.71489091998196e-08
+iteration 540 res= 2.5257605641697368e-08
+iteration 541 res= 2.5808061689101555e-08
+iteration 542 res= 2.4010167028157074e-08
+iteration 543 res= 2.4533436803896536e-08
+iteration 544 res= 2.2824337702395247e-08
+iteration 545 res= 2.332176389926172e-08
+iteration 546 res= 2.169707486591303e-08
+iteration 547 res= 2.216993386293683e-08
+iteration 548 res= 2.062548599986969e-08
+iteration 549 res= 2.107499113746573e-08
+iteration 550 res= 1.960682144297574e-08
+iteration 551 res= 2.003412613633461e-08
+iteration 552 res= 1.863846733595384e-08
+iteration 553 res= 1.904466803466666e-08
+iteration 554 res= 1.7717938914462987e-08
+iteration 555 res= 1.810407791597451e-08
+iteration 556 res= 1.6842874133276164e-08
+iteration 557 res= 1.720994225738524e-08
+iteration 558 res= 1.6011027605350997e-08
+iteration 559 res= 1.635996673662114e-08
+iteration 560 res= 1.522026484024154e-08
+iteration 561 res= 1.5551970344845307e-08
+iteration 562 res= 1.4468556767066967e-08
+iteration 563 res= 1.4783879790265702e-08
+iteration 564 res= 1.3753974527983358e-08
+iteration 565 res= 1.4053724178137518e-08
+iteration 566 res= 1.3074684528798829e-08
+iteration 567 res= 1.3359629953513065e-08
+iteration 568 res= 1.2428943734031961e-08
+iteration 569 res= 1.2699816093762213e-08
+iteration 570 res= 1.1815095194340904e-08
+iteration 571 res= 1.2072589538527745e-08
+iteration 572 res= 1.1231563794846636e-08
+iteration 573 res= 1.1476340845388954e-08
+iteration 574 res= 1.0676852213440688e-08
+iteration 575 res= 1.0909540060086053e-08
+iteration 576 res= 1.0149537078706607e-08
+iteration 577 res= 1.0370732790708593e-08
+iteration 578 res= 9.648265317596309e-09
+iteration 579 res= 9.85853647577431e-09
+iteration 580 res= 9.171750683489822e-09
+iteration 581 res= 9.371636836622493e-09
+iteration 582 res= 8.718770455729333e-09
+iteration 583 res= 8.908784505018746e-09
+iteration 584 res= 8.288162302158766e-09
+iteration 585 res= 8.468791817317806e-09
+iteration 586 res= 7.87882129661824e-09
+iteration 587 res= 8.05052976695821e-09
+iteration 588 res= 7.489697083740388e-09
+iteration 589 res= 7.652925107469133e-09
+iteration 590 res= 7.119791183773515e-09
+iteration 591 res= 7.2749575985557405e-09
+iteration 592 res= 6.768154430515881e-09
+iteration 593 res= 6.915657388196596e-09
+iteration 594 res= 6.433884535786848e-09
+iteration 595 res= 6.574102524035757e-09
+iteration 596 res= 6.116123774185534e-09
+iteration 597 res= 6.2494165876837496e-09
+iteration 598 res= 5.814056782196025e-09
+iteration 599 res= 5.940766445857234e-09
+iteration 600 res= 5.526908465991797e-09
+iteration 601 res= 5.64736011258677e-09
+iteration 602 res= 5.2539420125708e-09
+iteration 603 res= 5.368444717007244e-09
+iteration 604 res= 4.994456999117883e-09
+iteration 605 res= 5.103304571516332e-09
+iteration 606 res= 4.7477875957431944e-09
+iteration 607 res= 4.851259335344e-09
+iteration 608 res= 4.513300856984907e-09
+iteration 609 res= 4.611662268820762e-09
+iteration 610 res= 4.290395097692264e-09
+iteration 611 res= 4.3838985738652655e-09
+iteration 612 res= 4.078498349121558e-09
+iteration 613 res= 4.1673838164329166e-09
+iteration 614 res= 3.877066891283406e-09
+iteration 615 res= 3.961562426877617e-09
+iteration 616 res= 3.6855838577753927e-09
+iteration 617 res= 3.76590627437859e-09
+iteration 618 res= 3.5035579095201218e-09
+iteration 619 res= 3.579913311774293e-09
+iteration 620 res= 3.3305219740055337e-09
+iteration 621 res= 3.4031062873261557e-09
+iteration 622 res= 3.166032046792411e-09
+iteration 623 res= 3.2350315201065416e-09
+iteration 624 res= 3.0096660522138033e-09
+iteration 625 res= 3.075257735868607e-09
+iteration 626 res= 2.8610227603429404e-09
+iteration 627 res= 2.923374960410976e-09
+iteration 628 res= 2.7197207574506218e-09
+iteration 629 res= 2.7789934675976132e-09
+iteration 630 res= 2.58539746731031e-09
+iteration 631 res= 2.6417427793335563e-09
+iteration 632 res= 2.457708220839628e-09
+iteration 633 res= 2.511270714930479e-09
+iteration 634 res= 2.3363253716909865e-09
+iteration 635 res= 2.387242487422794e-09
+iteration 636 res= 2.220937455521986e-09
+iteration 637 res= 2.269339844515465e-09
+iteration 638 res= 2.1112483907883025e-09
+iteration 639 res= 2.1572602519592308e-09
+iteration 640 res= 2.006976719008319e-09
+iteration 641 res= 2.0507161172578146e-09
+iteration 642 res= 1.9078548825500506e-09
+iteration 643 res= 1.9494340517151698e-09
+iteration 644 res= 1.8136285380871835e-09
+iteration 645 res= 1.8531541689292016e-09
+iteration 646 res= 1.7240559039625827e-09
+iteration 647 res= 1.761629417931932e-09
+iteration 648 res= 1.6389071397846271e-09
+iteration 649 res= 1.6746249492649519e-09
+iteration 650 res= 1.5579637566644271e-09
+iteration 651 res= 1.5919175123635469e-09
+iteration 652 res= 1.4810180565806218e-09
+iteration 653 res= 1.5132948827031924e-09
+iteration 654 res= 1.4078725994331884e-09
+iteration 655 res= 1.4385553172385025e-09
+iteration 656 res= 1.3383396964187282e-09
+iteration 657 res= 1.3675070367373048e-09
+iteration 658 res= 1.2722409284272562e-09
+iteration 659 res= 1.2999677336815251e-09
+iteration 660 res= 1.209406688224717e-09
+iteration 661 res= 1.2357641044721813e-09
+iteration 662 res= 1.1496757452464792e-09
+iteration 663 res= 1.1747314047381234e-09
+iteration 664 res= 1.0928948318850833e-09
+iteration 665 res= 1.1167130266074749e-09
+iteration 666 res= 1.038918250210674e-09
+iteration 667 res= 1.0615600968570555e-09
+iteration 668 res= 9.87607498114971e-10
+iteration 669 res= 1.0091310949086567e-09
+iteration 670 res= 9.388309139194786e-10
+iteration 671 res= 9.592914896919583e-10
+iteration 672 res= 8.924633385360109e-10
+iteration 673 res= 9.119133944422888e-10
+iteration 674 res= 8.483857943126437e-10
+iteration 675 res= 8.668752385474531e-10
+iteration 676 res= 8.064851797410314e-10
+iteration 677 res= 8.240614556015946e-10
+iteration 678 res= 7.666539792417061e-10
+iteration 679 res= 7.833621868656547e-10
+iteration 680 res= 7.287899872826877e-10
+iteration 681 res= 7.44672999373511e-10
+iteration 682 res= 6.927960461234977e-10
+iteration 683 res= 7.078946179604703e-10
+iteration 684 res= 6.58579796511638e-10
+iteration 685 res= 6.729326705265068e-10
+iteration 686 res= 6.260534406918286e-10
+iteration 687 res= 6.396974458806003e-10
+iteration 688 res= 5.951335171198969e-10
+iteration 689 res= 6.081036635448131e-10
+iteration 690 res= 5.657406863032377e-10
+iteration 691 res= 5.780702549274283e-10
+iteration 692 res= 5.377995272183236e-10
+iteration 693 res= 5.495201553036462e-10
+iteration 694 res= 5.112383437828715e-10
+iteration 695 res= 5.223801060700719e-10
+iteration 696 res= 4.859889808860888e-10
+iteration 697 res= 4.965804667655817e-10
+iteration 698 res= 4.6198664950493167e-10
+iteration 699 res= 4.720550363762232e-10
+iteration 700 res= 4.3916976045763246e-10
+iteration 701 res= 4.4874088346562045e-10
+iteration 702 res= 4.1747976636791445e-10
+iteration 703 res= 4.265781846950086e-10
+iteration 704 res= 3.9686101143437545e-10
+iteration 705 res= 4.0551007131854113e-10
+iteration 706 res= 3.7726058861955906e-10
+iteration 707 res= 3.8548248325998474e-10
+iteration 708 res= 3.5862820389226123e-10
+iteration 709 res= 3.6644403039636705e-10
+iteration 710 res= 3.4091604717472303e-10
+iteration 711 res= 3.4834586069263457e-10
+iteration 712 res= 3.240786696635655e-10
+iteration 713 res= 3.3114153484896144e-10
+iteration 714 res= 3.0807286720967693e-10
+iteration 715 res= 3.147869071390532e-10
+iteration 716 res= 2.928575694578068e-10
+needed 716 iterations
+
    +
    +

    We observe that the gradient method converges similar fast as the Richardson iteration, but without the need of a good chosen relaxation parameter \(\alpha\).

    The comparison to Richardson iteration allows also to estimate the error reduction of the gradient method. Let @@ -485,6 +1239,24 @@

    The Gradient Method
    + + +
    +

    previous

    +

    20. The Richardson Iteration

    +
    +     + +
    +

    next

    +

    24. Preconditioning

    +
    + +
    @@ -513,7 +1285,7 @@

    The Gradient Method - © Copyright 2023. + © Copyright 2024.

    diff --git a/iterative/preconditioning.html b/iterative/preconditioning.html index 43954399..950d0e67 100644 --- a/iterative/preconditioning.html +++ b/iterative/preconditioning.html @@ -8,7 +8,7 @@ - Preconditioning — Introduction to Scientific Computing + 24. Preconditioning — Introduction to Scientific Computing @@ -32,7 +32,7 @@ - + @@ -62,6 +62,8 @@ + + @@ -123,8 +125,6 @@ - -
    @@ -174,7 +174,7 @@ @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -332,12 +364,12 @@

    Contents

    @@ -351,7 +383,7 @@

    Contents

    -

    Preconditioning#

    +

    24. Preconditioning#

    We call \(C\) a preconditioner to the matrix \(A\) if

    -

    Jacobi and Gauss Seidel Preconditioners#

    +

    25. Jacobi and Gauss Seidel Preconditioners#

    For a given residual \(r\), the Jacobi preconditioner computes

    \[ @@ -706,6 +2607,24 @@

    Jacobi and Gauss Seidel Preconditioners
    + + +
    +

    previous

    +

    23. The Gradient Method

    +
    +     + +
    +

    next

    +

    26. The Chebyshev Method

    +
    + +
    @@ -722,12 +2641,12 @@

    Jacobi and Gauss Seidel Preconditioners

    @@ -753,7 +2672,7 @@

    Jacobi and Gauss Seidel Preconditioners - © Copyright 2023. + © Copyright 2024.

    diff --git a/iterative/simple.html b/iterative/simple.html index 12c418d1..a01157b9 100644 --- a/iterative/simple.html +++ b/iterative/simple.html @@ -8,7 +8,7 @@ - Basic Iterative Methods — Introduction to Scientific Computing + 19. Basic Iterative Methods — Introduction to Scientific Computing @@ -32,7 +32,7 @@ - + @@ -62,6 +62,8 @@ + + @@ -123,8 +125,6 @@ - -
    @@ -174,7 +174,7 @@ @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -335,7 +367,7 @@

    Basic Iterative Methods

    -

    Basic Iterative Methods#

    +

    19. Basic Iterative Methods#

    In this chapter we learn

    +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -332,10 +364,10 @@

    Contents

    @@ -348,7 +380,7 @@

    Contents

    -

    Abstract Theory#

    +

    8. Abstract Theory#

    \(\DeclareMathOperator{\opdiv}{div}\) We have seen a couple of examples leading to variational problems of the form:

    Mixed variational problem: Find \(u \in V\) and \(p \in Q\) such that

    @@ -426,7 +458,7 @@

    Abstract Theory -

    Constrained minimization problem#

    +

    8.1. Constrained minimization problem#

    Now assume that \(a(.,.)\) is symmetric and coercive. Consider the constrained minimization problem

    \[ @@ -445,7 +477,7 @@

    Constrained minimization problem -

    Stokes equation within the abstract theory#

    +

    8.2. Stokes equation within the abstract theory#

    The Hilbert spaces are \(V := [H_0^1(\Omega)]^d\) and \(Q = L_2^0\), and the forms

    \begin{eqnarray*} @@ -467,7 +499,7 @@

    Stokes equation within the abstract theory\(a(.,.)\) is coercive on the whole space \(V\), and so also on the sub-space \(V_0\).

    -

    Dirichlet boundary conditions as mixed system#

    +

    8.3. Dirichlet boundary conditions as mixed system#

    \(V = H^1(\Omega)\), \(Q = H^{-1/2}(\partial \Omega)\) \begin{eqnarray*} a(u,v) &= & \int \nabla u \nabla v \ @@ -490,7 +522,7 @@

    Dirichlet boundary conditions as mixed system\(H_0^1\), the bilinear-form \(a(.,.)\) is continuous by Friedrichs’ inequality.

    -

    Mixed method for second order equation#

    +

    8.4. Mixed method for second order equation#

    Our Hilbert-spaces are \(\Sigma \times V = H(\operatorname{div}) \times L_2\), the forms are

    \begin{eqnarray*} a(\sigma,\tau) &= & \int \sigma \cdot \tau \ @@ -564,6 +596,24 @@

    Mixed method for second order equation
    + + +
    +

    previous

    +

    7. Mixed Methods for second order equations

    +
    +     + +
    +

    next

    +

    9. Abstract theory for mixed finite element methods

    +
    + +
    @@ -580,10 +630,10 @@

    Mixed method for second order equation @@ -608,7 +658,7 @@

    Mixed method for second order equation - © Copyright 2023. + © Copyright 2024.

    diff --git a/mixed/abstractfem.html b/mixed/abstractfem.html index a50b16f7..59b81ea2 100644 --- a/mixed/abstractfem.html +++ b/mixed/abstractfem.html @@ -8,7 +8,7 @@ - Abstract theory for mixed finite element methods — Introduction to Scientific Computing + 9. Abstract theory for mixed finite element methods — Introduction to Scientific Computing @@ -32,7 +32,7 @@ - + @@ -62,6 +62,8 @@ + + @@ -123,8 +125,6 @@ - -
    @@ -174,7 +174,7 @@ @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -332,9 +364,9 @@

    Contents

    @@ -347,7 +379,7 @@

    Contents

    -

    Abstract theory for mixed finite element methods#

    +

    9. Abstract theory for mixed finite element methods#

    In the finite element method we replace the spaces \(V\) and \(Q\) by sub-spaces \(V_h\) and \(Q_h\).

    We want to apply Brezzi’s theorem for the discrete setting: The (discrete) spaces are Hilbert-spaces, continuity of the forms is inherited to the discrete level. But, the LBB condition and kernel ellipticity must be verified on the discrete level case by case.

    We write the system as one big equation:

    @@ -367,7 +399,7 @@

    Abstract theory for mixed finite element methods -

    Finite element error estimates#

    +

    9.1. Finite element error estimates#

    We compare the finite element solution \(u_h, p_h\) to the interpolated true solution \(I_h u, I_h p\). The difference is a finite function, for which we apply the \(\inf-\sup\) condition. Then we use the Galerkin orthogonality, and continuity of the big-B:

    \begin{eqnarray*} | u_h - I_h u |_V + | p_h - I_h p |A @@ -388,7 +420,7 @@

    Finite element error estimates -

    Proving discrete LBB by the Fortin operator#

    +

    9.2. Proving discrete LBB by the Fortin operator#

    A Fortin-operator \(I^F : V \rightarrow V_h\) is a continuous operator (wrt to \(\| \cdot \|_V\)) which preservers the constraint in mean:

    \[ @@ -414,7 +446,7 @@

    Proving discrete LBB by the Fortin operator -

    Example: Finite elements for Stokes#

    +

    9.3. Example: Finite elements for Stokes#

    2D. Take the finite element pair \([P^2]^2 \times P^0\):

    We define the Fortin operator such that

    @@ -509,7 +559,7 @@

    Example: Finite elements for Stokes - © Copyright 2023. + © Copyright 2024.

    diff --git a/mixed/boundary.html b/mixed/boundary.html index d90f81ba..a969a6aa 100644 --- a/mixed/boundary.html +++ b/mixed/boundary.html @@ -8,7 +8,7 @@ - Boundary Conditions — Introduction to Scientific Computing + 6. Boundary Conditions — Introduction to Scientific Computing @@ -32,7 +32,7 @@ - + @@ -59,9 +59,14 @@ + + + + + @@ -123,8 +128,6 @@ - -
    @@ -174,7 +177,7 @@ @@ -184,6 +187,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -335,7 +370,7 @@

    Boundary Conditions

    -

    Boundary Conditions#

    +

    6. Boundary Conditions#

    Usually, Dirichlet boundary conditions (essential bc) are built into the space: find \(u \in H^1\) such that \(u = u_D\) on \(\Gamma_D\). Now, we want to pose the Dirichlet bc by an extra equation. For this, we start from the strong form

    \[ @@ -379,6 +414,11 @@

    Boundary Conditions +
    BaseWebGuiScene
+
    +
    +

    @@ -390,6 +430,14 @@

    Boundary Conditions +
    V.ndof = 283 Q.ndof = 40
+
    +
    +
    WARNING: kwarg 'orderface' is an undocumented flags option for class <class 'ngsolve.comp.H1'>, maybe there is a typo?
+
    +
    +

    @@ -419,6 +467,11 @@

    Boundary Conditions +
    BaseWebGuiScene
+
    +
    +

    Choosing the test-function \(v = 1\) in the first equation $\( @@ -436,6 +489,11 @@

    Boundary Conditions +
    4.999999999999997
+
    +
    +
    @@ -445,6 +503,16 @@

    Boundary Conditions +
    boundary parts:  2.24998
+ 1.88934
+ 0.250025
+ 0.610659
+
+sum:  4.9999999999999964
+
    +
    +

    @@ -478,6 +546,24 @@

    Boundary Conditions
    + + +
    +

    previous

    +

    5. Stokes Equation

    +
    +     + +
    +

    next

    +

    7. Mixed Methods for second order equations

    +
    + +
    @@ -506,7 +592,7 @@

    Boundary Conditions - © Copyright 2023. + © Copyright 2024.

    diff --git a/mixed/pardep.html b/mixed/pardep.html index bbd7f7e8..b94f3071 100644 --- a/mixed/pardep.html +++ b/mixed/pardep.html @@ -8,7 +8,7 @@ - Parameter Dependent Problems — Introduction to Scientific Computing + 10. Parameter Dependent Problems — Introduction to Scientific Computing @@ -32,7 +32,7 @@ - + @@ -62,6 +62,8 @@ + + @@ -123,8 +125,6 @@ - -
    @@ -174,7 +174,7 @@ @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -332,8 +364,8 @@

    Contents

    @@ -346,7 +378,7 @@

    Contents

    -

    Parameter Dependent Problems#

    +

    10. Parameter Dependent Problems#

    Many problems from mechanics are of the form

    \[ @@ -389,7 +421,7 @@

    Parameter Dependent Problems -

    Example: Dirichlet boundary condition by penalty#

    +

    10.1. Example: Dirichlet boundary condition by penalty#

    \[ \int \nabla u \nabla v + \frac{1}{\varepsilon} \int_{\Gamma_D} u v = \int f v @@ -401,7 +433,7 @@

    Example: Dirichlet boundary condition by penalty -

    Example: Nearly incompressible materials#

    +

    10.2. Example: Nearly incompressible materials#

    \[ \int 2 \mu \nabla^S u : \nabla^S v + \int \lambda \operatorname{div} u \operatorname{div} v = \int f v @@ -440,6 +472,24 @@

    Example: Nearly incompressible materials
    + + +
    +

    previous

    +

    9. Abstract theory for mixed finite element methods

    +
    +     + +
    +

    next

    +

    11. Stationary Transport Equation

    +
    + +
    @@ -456,8 +506,8 @@

    Example: Nearly incompressible materials

    @@ -482,7 +532,7 @@

    Example: Nearly incompressible materials - © Copyright 2023. + © Copyright 2024.

    diff --git a/mixed/secondorder.html b/mixed/secondorder.html index dc3fc573..3b46c721 100644 --- a/mixed/secondorder.html +++ b/mixed/secondorder.html @@ -8,7 +8,7 @@ - Mixed Methods for second order equations — Introduction to Scientific Computing + 7. Mixed Methods for second order equations — Introduction to Scientific Computing @@ -32,7 +32,7 @@ - + @@ -59,9 +59,14 @@ + + + + + @@ -123,8 +128,6 @@ - -
    @@ -174,7 +177,7 @@ @@ -184,6 +187,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -335,7 +370,7 @@

    Mixed Methods for second order equations

    @@ -413,6 +453,11 @@

    Mixed Methods for second order equations +
    BaseWebGuiScene
+
    +
    +

    The total outflow is in balance with the source:

    @@ -426,6 +471,11 @@

    Mixed Methods for second order equations +
    -0.9999999999999974
+
    +
    +

    @@ -459,6 +509,24 @@

    Mixed Methods for second order equations
    + + +
    +

    previous

    +

    6. Boundary Conditions

    +
    +     + +
    +

    next

    +

    8. Abstract Theory

    +
    + +
    @@ -487,7 +555,7 @@

    Mixed Methods for second order equations - © Copyright 2023. + © Copyright 2024.

    diff --git a/mixed/stokes.html b/mixed/stokes.html index 31275da4..9d1a1924 100644 --- a/mixed/stokes.html +++ b/mixed/stokes.html @@ -8,7 +8,7 @@ - Stokes Equation — Introduction to Scientific Computing + 5. Stokes Equation — Introduction to Scientific Computing @@ -32,7 +32,7 @@ - + @@ -59,9 +59,14 @@ + + + + + @@ -123,8 +128,6 @@ - -
    @@ -174,7 +177,7 @@ @@ -184,6 +187,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -332,8 +367,8 @@

    Contents

    @@ -346,7 +381,7 @@

    Contents

    -

    Stokes Equation#

    +

    5. Stokes Equation#

    The Stokes equation is the basic model for incompressible fluids. While several terms of the advanced Navier-Stokes equations are skipped, the Stokes equation allows to study the difficulty of incompressibility constraints.

    The unknown fields are

    \begin{eqnarray*} @@ -364,7 +399,7 @@

    Stokes Equation\(f : \Omega \rightarrow R^2\). The first equation models balance of momentum, the second one is the incompressibility constraint. In the simplest case, we set boundary conditions \(u = 0\) on \(\partial \Omega\)

    -

    Variational Formulation#

    +

    5.1. Variational Formulation#

    The weak form is : Find \(u \in V := [H_0^1(\Omega)]^d\) and \(p \in Q:= L_2^0(\Omega)\) such that

    \[\begin{split} @@ -376,7 +411,7 @@

    Variational FormulationThe pressure is defined up to a constant, and the divergence equation is satisfied for constant test-functions, anyway. Thus the space \(L_2^0\) of \(L_2\)-functions with zero mean value. For the beauty of the equation, we have substituted \(p\) by \(-p\).

    -

    Finite Element Spaces#

    +

    5.2. Finite Element Spaces#

    We discretize the Stokes equation by different finite element spaces. For the velocity we use continuous, \(H^1\)-conforming elements. For the pressure, which is a field in \(L_2\), discontinuous finite elements are natural. But, alternatively, one may also use continuous elements.

    We try different combinations of finite element spaces for \(V\) and \(Q\).

    @@ -389,6 +424,11 @@

    Finite Element Spaces

    +
    +
    BaseWebGuiScene
+
    +
    +

    Use continuous elements of order \(k=2\) for each velocity component, and piece-wise constants for pressure:

    @@ -425,6 +465,11 @@

    Finite Element Spaces

    +
    +
    BaseWebGuiScene
+
    +
    +

    Try a \(P^2 \times P^{1,dc}\) pairing:

    @@ -447,6 +492,11 @@

    Finite Element Spaces

    +
    +
    WARNING: kwarg 'orderinner' is an undocumented flags option for class <class 'ngsolve.comp.VectorH1'>, maybe there is a typo?
+
    +
    +

    The Taylor Hood element \(P^2 \times P^1\) with continuous pressure:

    @@ -493,6 +543,24 @@

    Finite Element Spaces
    + + +
    +

    previous

    +

    4. Iterative Solvers

    +
    +     + +
    +

    next

    +

    6. Boundary Conditions

    +
    + +
    @@ -509,8 +577,8 @@

    Finite Element Spaces

    @@ -535,7 +603,7 @@

    Finite Element Spaces - © Copyright 2023. + © Copyright 2024.

    diff --git a/mixedelasticity/dynamics.html b/mixedelasticity/dynamics.html index 9c8025ac..df9237b7 100644 --- a/mixedelasticity/dynamics.html +++ b/mixedelasticity/dynamics.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -403,7 +435,7 @@

    Non-linear dynamics - © Copyright 2023. + © Copyright 2024.

    diff --git a/mixedelasticity/hellingerreissner.html b/mixedelasticity/hellingerreissner.html index 287fc50e..e95b03cb 100644 --- a/mixedelasticity/hellingerreissner.html +++ b/mixedelasticity/hellingerreissner.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -469,7 +501,7 @@

    Dual mixed method - © Copyright 2023. + © Copyright 2024.

    diff --git a/mixedelasticity/reducedsymmetry.html b/mixedelasticity/reducedsymmetry.html index 5d16ab33..f596ffa6 100644 --- a/mixedelasticity/reducedsymmetry.html +++ b/mixedelasticity/reducedsymmetry.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -463,7 +495,7 @@

    \(\DeclareMathOperator{\opdiv}{di

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/mixedelasticity/tdnns.html b/mixedelasticity/tdnns.html index 88672e43..0a066d68 100644 --- a/mixedelasticity/tdnns.html +++ b/mixedelasticity/tdnns.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@ @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -580,7 +612,7 @@

    Error estimates: - © Copyright 2023. + © Copyright 2024.

    diff --git a/multigrid/Untitled.html b/multigrid/Untitled.html index 2356bae7..002d9b02 100644 --- a/multigrid/Untitled.html +++ b/multigrid/Untitled.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@     @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -414,7 +446,7 @@

    Contents

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/multigrid/Untitled1.html b/multigrid/Untitled1.html index 8ee7822b..7cc9015d 100644 --- a/multigrid/Untitled1.html +++ b/multigrid/Untitled1.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@ @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -414,7 +446,7 @@

    Contents

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/multigrid/algorithms.html b/multigrid/algorithms.html index 8133daeb..fbf0f720 100644 --- a/multigrid/algorithms.html +++ b/multigrid/algorithms.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@ @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -758,7 +790,7 @@

    Projection matrices from the finest level - © Copyright 2023. + © Copyright 2024.

    diff --git a/multigrid/analysisMG.html b/multigrid/analysisMG.html index 05f8b419..0558050c 100644 --- a/multigrid/analysisMG.html +++ b/multigrid/analysisMG.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -677,7 +709,7 @@

    Optimal convergence of the \(V\)<

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/multigrid/analysisML.html b/multigrid/analysisML.html index 6fc82b39..d2881630 100644 --- a/multigrid/analysisML.html +++ b/multigrid/analysisML.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@ @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -603,7 +635,7 @@

    Optimal analysis of the multi-level preconditioner - © Copyright 2023. + © Copyright 2024.

    diff --git a/multigrid/multilevel-extension.html b/multigrid/multilevel-extension.html index 760400ef..8743ea18 100644 --- a/multigrid/multilevel-extension.html +++ b/multigrid/multilevel-extension.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -601,7 +633,7 @@

    Extending boundary data - © Copyright 2023. + © Copyright 2024.

    diff --git a/multigrid/py-multigrid.html b/multigrid/py-multigrid.html index f1a69cd0..a16ab305 100644 --- a/multigrid/py-multigrid.html +++ b/multigrid/py-multigrid.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -758,7 +790,7 @@

    Projection matrices from the finest level - © Copyright 2023. + © Copyright 2024.

    diff --git a/plates/hhj-tdnns.html b/plates/hhj-tdnns.html index 9872e9dd..4940fdc7 100644 --- a/plates/hhj-tdnns.html +++ b/plates/hhj-tdnns.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -453,7 +485,7 @@

    Commuting diagram for \(H^1 - H(\

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/plates/hhj.html b/plates/hhj.html index 32c7528e..bec84032 100644 --- a/plates/hhj.html +++ b/plates/hhj.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@ @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -532,7 +564,7 @@

    Hellan-Herrmann-Johnson method - © Copyright 2023. + © Copyright 2024.

    diff --git a/plates/reissnermindlin.html b/plates/reissnermindlin.html index acfb80f4..ac45c70e 100644 --- a/plates/reissnermindlin.html +++ b/plates/reissnermindlin.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -484,7 +516,7 @@

    Reissner Mindlin Plates - © Copyright 2023. + © Copyright 2024.

    diff --git a/plates/shells.html b/plates/shells.html index f9fab23f..f558d67f 100644 --- a/plates/shells.html +++ b/plates/shells.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -588,7 +620,7 @@

    Nonlinear Shells - © Copyright 2023. + © Copyright 2024.

    diff --git a/plates/tdnnsplate.html b/plates/tdnnsplate.html index 39ff5ad8..6bcb29f0 100644 --- a/plates/tdnnsplate.html +++ b/plates/tdnnsplate.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -406,7 +438,7 @@

    3D-TDNNS Elasticity and Reissner Mindlin Plate elements - © Copyright 2023. + © Copyright 2024.

    diff --git a/preamble.html b/preamble.html index abac4bf8..a77e6d5a 100644 --- a/preamble.html +++ b/preamble.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@     @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -403,7 +435,7 @@

    Preamble

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/primal/Untitled.html b/primal/Untitled.html index b264bbd5..a2c8f365 100644 --- a/primal/Untitled.html +++ b/primal/Untitled.html @@ -32,7 +32,7 @@ - + @@ -59,7 +59,7 @@ - + @@ -177,7 +177,7 @@     @@ -187,6 +187,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -371,7 +403,7 @@

    mein erstes jupyter -
    BaseWebGuiScene
+
    BaseWebGuiScene
    @@ -436,7 +468,7 @@

    mein erstes jupyter - © Copyright 2023. + © Copyright 2024.

    diff --git a/primal/VO Okt 6.html b/primal/VO Okt 6.html index cfe99f9a..008df257 100644 --- a/primal/VO Okt 6.html +++ b/primal/VO Okt 6.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@     @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +

    @@ -400,7 +432,7 @@

    The Galerkin Method - © Copyright 2023. + © Copyright 2024.

    diff --git a/primal/boundary_conditions.html b/primal/boundary_conditions.html index 2fd5c37c..0f6defc3 100644 --- a/primal/boundary_conditions.html +++ b/primal/boundary_conditions.html @@ -32,7 +32,7 @@ - + @@ -59,7 +59,7 @@ - + @@ -177,7 +177,7 @@     @@ -187,6 +187,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -457,7 +489,7 @@

    2.4. Natural boundary conditions
    -
    +

    @@ -558,7 +590,7 @@

    2.5. Essential boundary conditions
    -
    +

    Now we compute the correction:

    @@ -667,7 +699,7 @@

    2.5. Essential boundary conditions - © Copyright 2023. + © Copyright 2024.

    diff --git a/primal/first_example.html b/primal/first_example.html index 51f875b0..21ad1f0c 100644 --- a/primal/first_example.html +++ b/primal/first_example.html @@ -32,7 +32,7 @@ - + @@ -59,14 +59,14 @@ - + - + @@ -177,7 +177,7 @@     @@ -187,6 +187,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +

    @@ -538,7 +570,7 @@

    1.4. Poisson equation in NGSolve:
    -
    +

    Number of vertices and elements:

    @@ -607,7 +639,7 @@

    1.4. Poisson equation in NGSolve:

    -
    +
    @@ -1501,7 +1533,7 @@

    1.4. Poisson equation in NGSolve:

    -
    +

    infinite loop for visualization of basis functions:

    @@ -2259,7 +2291,7 @@

    1.4. Poisson equation in NGSolve:1.4. Poisson equation in NGSolve:

    previous

    -

    Introduction to Scientific Computing

    +

    An Interactive Introduction to the Finite Element Method

     1.4. Poisson equation in NGSolve: - © Copyright 2023. + © Copyright 2024.

    diff --git a/primal/solvers.html b/primal/solvers.html index ee76067c..c7aaf5f4 100644 --- a/primal/solvers.html +++ b/primal/solvers.html @@ -32,7 +32,7 @@ - + @@ -57,12 +57,13 @@ - + + @@ -174,7 +175,7 @@     @@ -184,6 +185,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +

    @@ -361,7 +394,7 @@

    4. Iterative Solvers - +
    @@ -375,7 +408,7 @@

    4. Iterative Solvers -

    +
    @@ -609,7 +642,7 @@

    4. Iterative Solvers -

    +

    @@ -652,6 +685,15 @@

    4. Iterative Solvers3. Variable Coefficients

     + +
    +

    next

    +

    5. Stokes Equation

    +
    + +     @@ -680,7 +722,7 @@

    4. Iterative Solvers - © Copyright 2023. + © Copyright 2024.

    diff --git a/primal/subdomains.html b/primal/subdomains.html index cc40cf4e..86748513 100644 --- a/primal/subdomains.html +++ b/primal/subdomains.html @@ -32,7 +32,7 @@ - + @@ -59,7 +59,7 @@ - + @@ -177,7 +177,7 @@     @@ -187,6 +187,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -396,7 +428,7 @@

    3. Variable Coefficients - +
    @@ -407,7 +439,7 @@

    3. Variable Coefficients -

    +
    @@ -451,7 +483,7 @@

    3. Variable Coefficients -

    +

    use conductivity in bilinear-form:

    @@ -495,7 +527,7 @@

    3. Variable Coefficients -

    +
    @@ -504,7 +536,7 @@

    3. Variable Coefficients -

    +
    @@ -513,7 +545,7 @@

    3. Variable Coefficients -

    +

    @@ -593,7 +625,7 @@

    3. Variable Coefficients - © Copyright 2023. + © Copyright 2024.

    diff --git a/reports/DG/splitting.err.log b/reports/DG/splitting.err.log new file mode 100644 index 00000000..efdceed4 --- /dev/null +++ b/reports/DG/splitting.err.log @@ -0,0 +1,44 @@ +Traceback (most recent call last): + File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/nbclient/client.py", line 782, in _async_poll_for_reply + msg = await ensure_async(self.kc.shell_channel.get_msg(timeout=new_timeout)) + ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/jupyter_core/utils/__init__.py", line 198, in ensure_async + result = await obj + ^^^^^^^^^ + File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/jupyter_client/channels.py", line 315, in get_msg + raise Empty +_queue.Empty + +During handling of the above exception, another exception occurred: + +Traceback (most recent call last): + File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/jupyter_cache/executors/utils.py", line 58, in single_nb_execution + executenb( + File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/nbclient/client.py", line 1314, in execute + return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute() + ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/jupyter_core/utils/__init__.py", line 165, in wrapped + return loop.run_until_complete(inner) + ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/asyncio/base_events.py", line 685, in run_until_complete + return future.result() + ^^^^^^^^^^^^^^^ + File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/nbclient/client.py", line 709, in async_execute + await self.async_execute_cell( + File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/nbclient/client.py", line 1005, in async_execute_cell + exec_reply = await self.task_poll_for_reply + ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/nbclient/client.py", line 806, in _async_poll_for_reply + error_on_timeout_execute_reply = await self._async_handle_timeout(timeout, cell) + ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/nbclient/client.py", line 856, in _async_handle_timeout + raise CellTimeoutError.error_from_timeout_and_cell( +nbclient.exceptions.CellTimeoutError: A cell timed out while it was being executed, after 30 seconds. +The message was: Cell execution timed out. +Here is a preview of the cell contents: +------------------- +['gfu = GridFunction(fes)', 'gfu.components[0].Set(exp(-10**2*((x-0.5)*(x-0.5) +(y-0.75)*(y-0.75))))', 'scene = Draw(gfu.components[0], min=0, max=1, order=3, autoscale=False)', '', 'convu = gfu.vec.CreateVector()'] +... +[' w.data = inv * r', ' gfu.vec.data += tau*w', ' cnt += 1', ' if cnt % 10 == 0:', ' scene.Redraw()'] +------------------- + diff --git a/saddlepoint/augmented.html b/saddlepoint/augmented.html index f31976c9..067e19d7 100644 --- a/saddlepoint/augmented.html +++ b/saddlepoint/augmented.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@     @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -394,7 +426,7 @@

    Augmented Lagrangian - © Copyright 2023. + © Copyright 2024.

    diff --git a/saddlepoint/bramblepasciak.html b/saddlepoint/bramblepasciak.html index 4108b0f8..d2b1c336 100644 --- a/saddlepoint/bramblepasciak.html +++ b/saddlepoint/bramblepasciak.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -580,7 +612,7 @@

    The Bramble-Pasciak Transformation - © Copyright 2023. + © Copyright 2024.

    diff --git a/saddlepoint/bramblepasciak_save.html b/saddlepoint/bramblepasciak_save.html index fb251934..a277a879 100644 --- a/saddlepoint/bramblepasciak_save.html +++ b/saddlepoint/bramblepasciak_save.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -689,7 +721,7 @@

    The saddle point system

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/saddlepoint/fewconstraints.html b/saddlepoint/fewconstraints.html index 73a622be..99d0abc6 100644 --- a/saddlepoint/fewconstraints.html +++ b/saddlepoint/fewconstraints.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@ @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -677,7 +709,7 @@

    Projected preconditioner - © Copyright 2023. + © Copyright 2024.

    diff --git a/saddlepoint/parameterdependent.html b/saddlepoint/parameterdependent.html index 756c5737..b4a50409 100644 --- a/saddlepoint/parameterdependent.html +++ b/saddlepoint/parameterdependent.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -874,7 +906,7 @@

    Two-level analysis for Maxwell equations - © Copyright 2023. + © Copyright 2024.

    diff --git a/saddlepoint/structure.html b/saddlepoint/structure.html index 4d76873f..898feda4 100644 --- a/saddlepoint/structure.html +++ b/saddlepoint/structure.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -806,7 +838,7 @@

    Block-preconditioning - © Copyright 2023. + © Copyright 2024.

    diff --git a/saddlepoint/useBP.html b/saddlepoint/useBP.html index 71768ed9..45770da7 100644 --- a/saddlepoint/useBP.html +++ b/saddlepoint/useBP.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@     @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -521,7 +553,7 @@

    Contents

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/search.html b/search.html index 511169ca..cac4c31f 100644 --- a/search.html +++ b/search.html @@ -183,6 +183,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -341,7 +373,7 @@

    Search

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/searchindex.js b/searchindex.js index 729c8f2f..2bcab0fc 100644 --- a/searchindex.js +++ b/searchindex.js @@ -1 +1 @@ -Search.setIndex({"docnames": ["DG/Nitsche", "DG/elliptic", "DG/elliptic_stdDG", "DG/fourthorder", "DG/instationary", "DG/splitting", "DG/stationary", "DG/stokes", "MPIparallel/PETSc_interface", "MPIparallel/distmesh", "MPIparallel/hdivnstsimple/NavierStokesMT", "MPIparallel/hdivnstsimple/TestConvert", "MPIparallel/hdivnstsimple/veclaplace", "MPIparallel/hdivnstsimple/veclaplace-auxiliary", "MPIparallel/intro", "MPIparallel/paralleliteration", "MPIparallel/parallelstokes", "MPIparallel/petsc", "MPIparallel/vectypes", "README", "domaindecomposition/BDDC", "domaindecomposition/FETI", "domaindecomposition/FETI-DP", "domaindecomposition/introduction", "domaindecomposition/traces", "ex20210609", "helmholtz/absorbing", "helmholtz/grating", "iFEM", "intro", "iterative/Chebyshev", "iterative/Richardson", "iterative/conjugategradients", "iterative/gradientmethod", "iterative/preconditioning", "iterative/simple", "mixed/abstract", "mixed/abstractfem", "mixed/boundary", "mixed/pardep", "mixed/secondorder", "mixed/stokes", "mixedelasticity/dynamics", "mixedelasticity/hellingerreissner", "mixedelasticity/reducedsymmetry", "mixedelasticity/tdnns", "multigrid/Untitled", "multigrid/Untitled1", "multigrid/algorithms", "multigrid/analysisMG", "multigrid/analysisML", "multigrid/multilevel-extension", "multigrid/py-multigrid", "plates/hhj", "plates/hhj-tdnns", "plates/reissnermindlin", "plates/shells", "plates/tdnnsplate", "preamble", "primal/Untitled", "primal/VO Okt 6", "primal/boundary_conditions", "primal/first_example", "primal/solvers", "primal/subdomains", "saddlepoint/augmented", "saddlepoint/bramblepasciak", "saddlepoint/bramblepasciak_save", "saddlepoint/fewconstraints", "saddlepoint/parameterdependent", "saddlepoint/structure", "saddlepoint/useBP", "secondorder/erroranalysis", "secondorder/erroranalysisl2h1", "secondorder/finiteelements", "secondorder/framework", "secondorder/hdiv", "secondorder/hybridization", "sobolev/Friedrichs", "sobolev/Untitled", "sobolev/Untitled1", "sobolev/Untitled2", "sobolev/poincare", "sobolev/trace", "subspacecorrection/Untitled", "subspacecorrection/asm", "subspacecorrection/examples", "subspacecorrection/exercise20210324", "subspacecorrection/minimaldd", "subspacecorrection/overlapping", "timedependent/ODEs/runge_kutta", "timedependent/ODEs/singlestep", "timedependent/intro/Untitled", "timedependent/intro/heatequation", "timedependent/intro/waveequation", "timedependent/parabolic/exponential", "timedependent/waves/Maxwell-DG", "timedependent/waves/Untitled", "timedependent/waves/lts", "timedependent/waves/ringresonator", "timedependent/waves/testpml/ringresonator", "timedependent/waves/wave-leapfrogDG"], "filenames": ["DG/Nitsche.ipynb", "DG/elliptic.ipynb", "DG/elliptic_stdDG.ipynb", "DG/fourthorder.ipynb", "DG/instationary.ipynb", "DG/splitting.ipynb", "DG/stationary.ipynb", "DG/stokes.ipynb", "MPIparallel/PETSc_interface.ipynb", "MPIparallel/distmesh.ipynb", "MPIparallel/hdivnstsimple/NavierStokesMT.ipynb", "MPIparallel/hdivnstsimple/TestConvert.ipynb", "MPIparallel/hdivnstsimple/veclaplace.ipynb", "MPIparallel/hdivnstsimple/veclaplace-auxiliary.ipynb", "MPIparallel/intro.ipynb", "MPIparallel/paralleliteration.ipynb", "MPIparallel/parallelstokes.ipynb", "MPIparallel/petsc.ipynb", "MPIparallel/vectypes.ipynb", "README.md", "domaindecomposition/BDDC.ipynb", "domaindecomposition/FETI.ipynb", "domaindecomposition/FETI-DP.ipynb", "domaindecomposition/introduction.ipynb", "domaindecomposition/traces.ipynb", "ex20210609.ipynb", "helmholtz/absorbing.ipynb", "helmholtz/grating.ipynb", "iFEM.ipynb", "intro.md", "iterative/Chebyshev.ipynb", "iterative/Richardson.ipynb", "iterative/conjugategradients.ipynb", "iterative/gradientmethod.ipynb", "iterative/preconditioning.ipynb", "iterative/simple.ipynb", "mixed/abstract.ipynb", "mixed/abstractfem.ipynb", "mixed/boundary.ipynb", "mixed/pardep.ipynb", "mixed/secondorder.ipynb", "mixed/stokes.ipynb", "mixedelasticity/dynamics.ipynb", "mixedelasticity/hellingerreissner.ipynb", "mixedelasticity/reducedsymmetry.ipynb", "mixedelasticity/tdnns.ipynb", "multigrid/Untitled.ipynb", "multigrid/Untitled1.ipynb", "multigrid/algorithms.ipynb", "multigrid/analysisMG.ipynb", "multigrid/analysisML.ipynb", "multigrid/multilevel-extension.ipynb", "multigrid/py-multigrid.ipynb", "plates/hhj.ipynb", "plates/hhj-tdnns.ipynb", "plates/reissnermindlin.ipynb", "plates/shells.ipynb", "plates/tdnnsplate.ipynb", "preamble.ipynb", "primal/Untitled.ipynb", "primal/VO Okt 6.ipynb", "primal/boundary_conditions.ipynb", "primal/first_example.ipynb", "primal/solvers.ipynb", "primal/subdomains.ipynb", "saddlepoint/augmented.ipynb", "saddlepoint/bramblepasciak.ipynb", "saddlepoint/bramblepasciak_save.ipynb", "saddlepoint/fewconstraints.ipynb", "saddlepoint/parameterdependent.ipynb", "saddlepoint/structure.ipynb", "saddlepoint/useBP.ipynb", "secondorder/erroranalysis.ipynb", "secondorder/erroranalysisl2h1.ipynb", "secondorder/finiteelements.ipynb", "secondorder/framework.ipynb", "secondorder/hdiv.ipynb", "secondorder/hybridization.ipynb", "sobolev/Friedrichs.ipynb", "sobolev/Untitled.ipynb", "sobolev/Untitled1.ipynb", "sobolev/Untitled2.ipynb", "sobolev/poincare.ipynb", "sobolev/trace.ipynb", "subspacecorrection/Untitled.ipynb", "subspacecorrection/asm.ipynb", "subspacecorrection/examples.ipynb", "subspacecorrection/exercise20210324.ipynb", "subspacecorrection/minimaldd.ipynb", "subspacecorrection/overlapping.ipynb", "timedependent/ODEs/runge_kutta.ipynb", "timedependent/ODEs/singlestep.ipynb", "timedependent/intro/Untitled.ipynb", "timedependent/intro/heatequation.ipynb", "timedependent/intro/waveequation.ipynb", "timedependent/parabolic/exponential.ipynb", "timedependent/waves/Maxwell-DG.ipynb", "timedependent/waves/Untitled.ipynb", "timedependent/waves/lts.ipynb", "timedependent/waves/ringresonator.ipynb", "timedependent/waves/testpml/ringresonator.ipynb", "timedependent/waves/wave-leapfrogDG.ipynb"], "titles": ["Nitsche\u2019s Method for boundary and interface conditions", "Hybrid DG for elliptic equations", "DG - Methods for elliptic problems", "Fourth Order Equation", "Instationary Transport Equation", "Splitting Methods for the time-dependent convection diffusion equation", "Stationary Transport Equation", "H(div)-conforming Stokes", "NGSolve - PETSc interface", "Distributed Meshes and Spaces", "<no title>", "<no title>", "<no title>", "<no title>", "Introduction to MPI with mpi4py", "Iteration methods in parallel", "Solving Stokes in parallel", "Using PETSc", "Consistent and Distributed Vectors", "This repository contains an interactive introduction to the Finite Element Method", "BDDC - Preconditioner", "FETI methods", "FETI-DP", "Introduction to Non-overlapping Domain Decomposition", "Traces spaces", "Implement a parallel BDDC preconditioner", "Helmholtz Equation", "Grating", "An Interactive Introduction to the Finite Element Method", "An Interactive Introduction to the Finite Element Method", "The Chebyshev Method", "The Richardson Iteration", "Conjugate Gradients", "The Gradient Method", "Preconditioning", "Basic Iterative Methods", "Abstract Theory", "Abstract theory for mixed finite element methods", "Boundary Conditions", "Parameter Dependent Problems", "Mixed Methods for second order equations", "Stokes Equation", "Non-linear dynamics", "\\(\\DeclareMathOperator{\\opdiv}{div}\\)\nHellinger Reissner mixed formulation", "\\(\\DeclareMathOperator{\\opdiv}{div}\\)\n\\(\\DeclareMathOperator{\\opcurl}{curl}\\)\n\\(\\DeclareMathOperator{\\eps}{\\varepsilon}\\)\nReduced symmetry methods", "Tangential displacement normal normal stress continuous finite elements", "<no title>", "<no title>", "Multigrid and Multilevel Methods", "Analysis of the Multigrid Iteration", "Analysis of the multi-level preconditioner", "Multi-level Extension", "Multigrid and Multilevel Methods", "The Hellan Herrmann Johnson Method for Kirchhoff plates", "Relationship between HHJ and TDNNS", "Reissner Mindlin Plates", "Nonlinear Shells", "3D-TDNNS Elasticity and Reissner Mindlin Plate elements", "Preamble", "mein erstes jupyter", "The Galerkin Method", "2. Boundary Conditions", "1. Solving the Poisson Equation", "4. Iterative Solvers", "3. Variable Coefficients", "Augmented Lagrangian", "The Bramble-Pasciak Transformation", "The Bramble-Pasciak Transformation", "A Small Number of Constraints", "Parameter Dependent Problems", "Structure of Saddle-point Problems", "<no title>", "Finite Element Error Analysis", "Error Analysis in \\(L_2 \\times H^1\\)", "Finite Elements in \\(H(\\operatorname{div})\\)", "Application of the abstract theory", "The function space \\(H(\\operatorname{div})\\)", "Hybridization Techniques", "Friedrichs\u2019 Inequality", "<no title>", "<no title>", "Friedrichs\u2019 Inequality", "The Poincar\u00e9 inequality", "The Trace Inequality", "<no title>", "Additive Schwarz Methods", "Same Examples of ASM preconditioners", "Develop robust preconditioners", "Domain Decomposition with minimal overlap", "Overlapping Domain Decomposition Methods", "Runge Kutta Methods", "Single-step methods", "<no title>", "Heat Equation", "Wave Equation", "Exponential Integrators for Parabolic Equations", "<no title>", "<no title>", "Mass-lumping and Local time-stepping", "Nano-optics: A ring-resonator", "Nano-optics: A ring-resonator", "Discontinuous Galerkin for the Wave Equation"], "terms": {"from": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 45, 49, 50, 51, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 82, 83, 85, 86, 88, 89, 90, 91, 93, 94, 95, 96, 97, 98, 99, 100, 101], "netgen": [0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 16, 20, 21, 22, 25, 26, 27, 30, 31, 32, 33, 34, 38, 40, 41, 45, 48, 49, 51, 52, 53, 55, 56, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 74, 75, 77, 86, 88, 89, 93, 94, 95, 98, 101], "occ": [0, 10, 12, 13, 16, 59, 61, 62, 63, 64, 71, 93, 94, 95, 98, 101], "import": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 25, 26, 27, 30, 31, 32, 33, 34, 37, 38, 40, 41, 45, 48, 49, 51, 52, 53, 55, 56, 59, 61, 62, 63, 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"the-upper-bound-by-the-overlap"]], "Same Examples of ASM preconditioners": [[86, "same-examples-of-asm-preconditioners"]], "Diagonal preconditioner for L_2-norm": [[86, "diagonal-preconditioner-for-l-2-norm"]], "Diagonal preconditioner for the H^1 norm": [[86, "diagonal-preconditioner-for-the-h-1-norm"]], "H^1-norm with small L_2-term": [[86, "h-1-norm-with-small-l-2-term"]], "Develop robust preconditioners": [[87, "develop-robust-preconditioners"]], "Team A:": [[87, "team-a"]], "Team B:": [[87, "team-b"]], "Domain Decomposition with minimal overlap": [[88, "domain-decomposition-with-minimal-overlap"]], "Analysis of the method": [[88, "analysis-of-the-method"]], "Adding a coarse grid space": [[88, "adding-a-coarse-grid-space"]], "Graph-based mesh partitioning": [[88, "graph-based-mesh-partitioning"]], "Overlapping Domain Decomposition Methods": [[89, "overlapping-domain-decomposition-methods"]], "Experiments with overlapping DD": [[89, "experiments-with-overlapping-dd"]], "Analysis of 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"newmark-time-stepping-method"]], "Exponential Integrators for Parabolic Equations": [[95, "exponential-integrators-for-parabolic-equations"]], "Mass-lumping and Local time-stepping": [[98, "mass-lumping-and-local-time-stepping"]], "The Verlet method:": [[98, "the-verlet-method"]], "Geometry with local details:": [[98, "geometry-with-local-details"]], "Nano-optics: A ring-resonator": [[99, "nano-optics-a-ring-resonator"], [100, "nano-optics-a-ring-resonator"]], "Discontinuous Galerkin for the Wave Equation": [[101, "discontinuous-galerkin-for-the-wave-equation"]], "Testing the differential operators": [[101, "testing-the-differential-operators"]], "Efficient implementation:": [[101, "efficient-implementation"]], "Solving the wave equation:": [[101, "solving-the-wave-equation"]], "Eigenvalues of the discretized Laplace-operator": [[101, "eigenvalues-of-the-discretized-laplace-operator"]]}, "indexentries": {}}) \ No newline at end of file +Search.setIndex({"docnames": ["DG/Nitsche", 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"example-dirichlet-boundary-condition-by-penalty"]], "Example: Nearly incompressible materials": [[39, "example-nearly-incompressible-materials"]], "Mixed Methods for second order equations": [[40, "mixed-methods-for-second-order-equations"]], "Stokes Equation": [[41, "stokes-equation"]], "Variational Formulation": [[41, "variational-formulation"]], "Finite Element Spaces": [[41, "finite-element-spaces"]], "Non-linear dynamics": [[42, "non-linear-dynamics"]], "\\DeclareMathOperator{\\opdiv}{div}\nHellinger Reissner mixed formulation": [[43, "declaremathoperator-opdiv-div-hellinger-reissner-mixed-formulation"]], "Primal mixed method:": [[43, "primal-mixed-method"]], "Dual mixed method": [[43, "dual-mixed-method"]], "\\DeclareMathOperator{\\opdiv}{div}\n\\DeclareMathOperator{\\opcurl}{curl}\n\\DeclareMathOperator{\\eps}{\\varepsilon}\nReduced symmetry methods": [[44, "declaremathoperator-opdiv-div-declaremathoperator-opcurl-curl-declaremathoperator-eps-varepsilon-reduced-symmetry-methods"]], "Tangential displacement normal normal stress continuous finite elements": [[45, "tangential-displacement-normal-normal-stress-continuous-finite-elements"]], "The divergence of nn-continuous piece-wise smooth functions:": [[45, "the-divergence-of-nn-continuous-piece-wise-smooth-functions"]], "TDNNS Variational formulation:": [[45, "tdnns-variational-formulation"]], "Error estimates:": [[45, "error-estimates"]], "Multigrid and Multilevel Methods": [[48, "multigrid-and-multilevel-methods"], [52, "multigrid-and-multilevel-methods"]], "Multilevel preconditioner": [[48, "multilevel-preconditioner"], [52, "multilevel-preconditioner"]], "Multigrid Preconditioning": [[48, "multigrid-preconditioning"], [52, "multigrid-preconditioning"]], "Projection matrices from the finest level": [[48, "projection-matrices-from-the-finest-level"], [52, "projection-matrices-from-the-finest-level"]], "Analysis of the Multigrid Iteration": [[49, "analysis-of-the-multigrid-iteration"]], "The Algorithm": [[49, "the-algorithm"]], "The Approximation Property": [[49, "the-approximation-property"]], "The Smothing Property": [[49, "the-smothing-property"]], "Optimal convergence of the V-cycle": [[49, "optimal-convergence-of-the-v-cycle"]], "Analysis of the multi-level preconditioner": [[50, "analysis-of-the-multi-level-preconditioner"]], "Nearly optimal analysis of the ML - preconditioner": [[50, "nearly-optimal-analysis-of-the-ml-preconditioner"]], "Optimal analysis of the multi-level preconditioner": [[50, "optimal-analysis-of-the-multi-level-preconditioner"]], "Multi-level Extension": [[51, "multi-level-extension"]], "Efficiently computable multi-level decomposition": [[51, "efficiently-computable-multi-level-decomposition"]], "Algorithm": [[51, "algorithm"]], "Extending boundary data": [[51, "extending-boundary-data"]], "The Hellan Herrmann Johnson Method for Kirchhoff plates": [[53, "the-hellan-herrmann-johnson-method-for-kirchhoff-plates"]], "Kirchhoff Plate equation": [[53, "kirchhoff-plate-equation"]], "Hellan-Herrmann-Johnson method": [[53, "hellan-herrmann-johnson-method"]], "Relationship between HHJ and TDNNS": [[54, "relationship-between-hhj-and-tdnns"]], "Commuting diagram for H^1 - H(\\opcurl)": [[54, "commuting-diagram-for-h-1-h-opcurl"]], "Reissner Mindlin Plates": [[55, "reissner-mindlin-plates"]], "Nonlinear Shells": [[56, "nonlinear-shells"]], "3D-TDNNS Elasticity and Reissner Mindlin Plate elements": [[57, "d-tdnns-elasticity-and-reissner-mindlin-plate-elements"]], "Preamble": [[58, "preamble"]], "mein erstes jupyter": [[59, "mein-erstes-jupyter"]], "Dirichlet boundary conditions": [[61, "dirichlet-boundary-conditions"]], "Neumann boundary conditions": [[61, "neumann-boundary-conditions"]], "Robin boundary conditions": [[61, "robin-boundary-conditions"]], "Natural boundary conditions": [[61, "natural-boundary-conditions"]], "Essential boundary conditions": [[61, "essential-boundary-conditions"]], "Solving the Poisson Equation": [[62, "solving-the-poisson-equation"]], "Weak formulation": [[62, "weak-formulation"]], "The Sobolev space H^1, linear and bilinear forms": [[62, "the-sobolev-space-h-1-linear-and-bilinear-forms"]], "The Finite Element Method": [[62, "the-finite-element-method"]], "Poisson equation in NGSolve:": [[62, "poisson-equation-in-ngsolve"]], "Variable Coefficients": [[64, "variable-coefficients"]], "Augmented Lagrangian": [[65, "augmented-lagrangian"]], "The Bramble-Pasciak Transformation": [[66, "the-bramble-pasciak-transformation"], [67, "the-bramble-pasciak-transformation"]], "The saddle point system\n$$\n\\left( \\begin{array}{ccc} A & B^T \\ B & 0 \\end{array} \\right)\n\\left( \\begin{array}{c} u \\ p \\end{array} \\right)": [[67, "the-saddle-point-system-left-begin-array-ccc-a-b-t-b-0-end-array-right-left-begin-array-c-u-p-end-array-right"]], "A Small Number of Constraints": [[68, "a-small-number-of-constraints"]], "Projected preconditioner": [[68, "projected-preconditioner"]], "Dirichlet boundary conditions by penalty:": [[69, "dirichlet-boundary-conditions-by-penalty"]], "Penalty formulation for the flux:": [[69, "penalty-formulation-for-the-flux"]], "Maxwell equations:": [[69, "maxwell-equations"]], "Penalty formulation for the Stokes equation:": [[69, "penalty-formulation-for-the-stokes-equation"]], "Robust two-level methods for parameter dependent problems": [[69, "robust-two-level-methods-for-parameter-dependent-problems"]], "Robust smoothers": [[69, "robust-smoothers"]], "Robust coarse-grid correction": [[69, "robust-coarse-grid-correction"]], "Two-level analysis for Maxwell equations": [[69, "two-level-analysis-for-maxwell-equations"]], "Structure of Saddle-point Problems": [[70, "structure-of-saddle-point-problems"]], "Constrained minimization problems": [[70, "constrained-minimization-problems"]], "Examples": [[70, "examples"], [90, "examples"]], "Schur complement iteration": [[70, "schur-complement-iteration"]], "Block-preconditioning": [[70, "block-preconditioning"]], "Finite Element Error Analysis": [[72, "finite-element-error-analysis"]], "Local post-processing": [[72, "local-post-processing"]], "Error Analysis in L_2 \\times H^1": [[73, "error-analysis-in-l-2-times-h-1"]], "Finite Elements in H(\\operatorname{div})": [[74, "finite-elements-in-h-operatorname-div"]], "Piola Transformation": [[74, "piola-transformation"]], "Application of the abstract theory": [[75, "application-of-the-abstract-theory"]], "Dual mixed formulation": [[75, "dual-mixed-formulation"]], "Primal mixed formulation": [[75, "primal-mixed-formulation"]], "The function space H(\\operatorname{div})": [[76, "the-function-space-h-operatorname-div"]], "Normal-trace of functions in H(\\opdiv)": [[76, "normal-trace-of-functions-in-h-opdiv"]], "H(\\opdiv) on sub-domains": [[76, "h-opdiv-on-sub-domains"]], "Hybridization Techniques": [[77, "hybridization-techniques"]], "Friedrichs\u2019 Inequality": [[78, "friedrichs-inequality"], [81, "friedrichs-inequality"]], "The Poincar\u00e9 inequality": [[82, "the-poincare-inequality"]], "The Trace Inequality": [[83, "the-trace-inequality"]], "Numerical verification": [[83, "numerical-verification"]], "Proof of the trace inequality:": [[83, "proof-of-the-trace-inequality"]], "Additive Schwarz Methods": [[85, "additive-schwarz-methods"]], "Block-Jacobi and general additive Schwarz preconditioners": [[85, "block-jacobi-and-general-additive-schwarz-preconditioners"]], "Sub-spaces of {\\mathbb R}^N and of Hilbert-spaces": [[85, "sub-spaces-of-mathbb-r-n-and-of-hilbert-spaces"]], "The Additive Schwarz Lemma": [[85, "the-additive-schwarz-lemma"]], "The upper-bound by the overlap": [[85, "the-upper-bound-by-the-overlap"]], "Same Examples of ASM preconditioners": [[86, "same-examples-of-asm-preconditioners"]], "Diagonal preconditioner for L_2-norm": [[86, "diagonal-preconditioner-for-l-2-norm"]], "Diagonal preconditioner for the H^1 norm": [[86, "diagonal-preconditioner-for-the-h-1-norm"]], "H^1-norm with small L_2-term": [[86, "h-1-norm-with-small-l-2-term"]], "Develop robust preconditioners": [[87, "develop-robust-preconditioners"]], "Team A:": [[87, "team-a"]], "Team B:": [[87, "team-b"]], "Domain Decomposition with minimal overlap": [[88, "domain-decomposition-with-minimal-overlap"]], "Analysis of the method": [[88, "analysis-of-the-method"]], "Adding a coarse grid space": [[88, "adding-a-coarse-grid-space"]], "Graph-based mesh partitioning": [[88, "graph-based-mesh-partitioning"]], "Overlapping Domain Decomposition Methods": [[89, "overlapping-domain-decomposition-methods"]], "Experiments with overlapping DD": [[89, "experiments-with-overlapping-dd"]], "Analysis of the DD preconditioner": [[89, "analysis-of-the-dd-preconditioner"]], "Overlapping DD Methods with coarse grid": [[89, "overlapping-dd-methods-with-coarse-grid"]], "Analysis of the 2-level method:": [[89, "analysis-of-the-2-level-method"]], "Comparison to DD with minimal overlap": [[89, "comparison-to-dd-with-minimal-overlap"]], "Runge Kutta Methods": [[90, "runge-kutta-methods"]], "Butcher tableaus of simple methods": [[90, "butcher-tableaus-of-simple-methods"]], "Convergence of Runge Kutta methods": [[90, "convergence-of-runge-kutta-methods"]], "Explicit methods:": [[90, "explicit-methods"]], "Diagonally implicit Runge-Kutta methods:": [[90, "diagonally-implicit-runge-kutta-methods"]], "Exercise:": [[90, "exercise"], [101, "exercise"]], "Single-step methods": [[91, "single-step-methods"]], "Simple methods based on the integral equation": [[91, "simple-methods-based-on-the-integral-equation"]], "Explicit Euler method (EE)": [[91, "explicit-euler-method-ee"]], "Implicit Euler method (IE)": [[91, "implicit-euler-method-ie"]], "Trapezoidal method": [[91, "trapezoidal-method"]], "Explicit mid-point rule (=improved Euler method = RK2 method)": [[91, "explicit-mid-point-rule-improved-euler-method-rk2-method"]], "Linear stability classification": [[91, "linear-stability-classification"]], "Stability function": [[91, "stability-function"]], "Stability conditions": [[91, "stability-conditions"]], "Single step methods and parabolic equations": [[91, "single-step-methods-and-parabolic-equations"]], "Single step methods and hyperbolic equations": [[91, "single-step-methods-and-hyperbolic-equations"]], "Exercises:": [[91, "exercises"]], "Heat Equation": [[93, "heat-equation"]], "Variational formulation in space": [[93, "variational-formulation-in-space"]], "Galerkin method in space": [[93, "galerkin-method-in-space"]], "Implicit Euler time-stepping": [[93, "implicit-euler-time-stepping"]], "Wave Equation": [[94, "wave-equation"]], "Newmark time-stepping method": [[94, "newmark-time-stepping-method"]], "Exponential Integrators for Parabolic Equations": [[95, "exponential-integrators-for-parabolic-equations"]], "Mass-lumping and Local time-stepping": [[98, "mass-lumping-and-local-time-stepping"]], "The Verlet method:": [[98, "the-verlet-method"]], "Geometry with local details:": [[98, "geometry-with-local-details"]], "Nano-optics: A ring-resonator": [[99, "nano-optics-a-ring-resonator"], [100, "nano-optics-a-ring-resonator"]], "Discontinuous Galerkin for the Wave Equation": [[101, "discontinuous-galerkin-for-the-wave-equation"]], "Testing the differential operators": [[101, "testing-the-differential-operators"]], "Efficient implementation:": [[101, "efficient-implementation"]], "Solving the wave equation:": [[101, "solving-the-wave-equation"]], "Eigenvalues of the discretized Laplace-operator": [[101, "eigenvalues-of-the-discretized-laplace-operator"]]}, "indexentries": {}}) \ No newline at end of file diff --git a/secondorder/erroranalysis.html b/secondorder/erroranalysis.html index 2badd30c..8d809018 100644 --- a/secondorder/erroranalysis.html +++ b/secondorder/erroranalysis.html @@ -32,7 +32,7 @@ - + @@ -59,9 +59,14 @@ + + + + + @@ -123,8 +128,6 @@ - -
    @@ -174,7 +177,7 @@ @@ -184,6 +187,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -453,6 +488,8 @@

    Local post-processing +
    +

    Now do the postprocessing. This requires to solve decoupled problems on every element, what is cheap.

    @@ -477,6 +514,11 @@

    Local post-processing

    +
    +
    BaseWebGuiScene
+
    +
    +

    @@ -511,6 +553,24 @@

    Local post-processing
    + + +
    +

    previous

    +

    Finite Elements in \(H(\operatorname{div})\)

    +
    +     + +
    +

    next

    +

    Error Analysis in \(L_2 \times H^1\)

    +
    + +
    @@ -552,7 +612,7 @@

    Local post-processing - © Copyright 2023. + © Copyright 2024.

    diff --git a/secondorder/erroranalysisl2h1.html b/secondorder/erroranalysisl2h1.html index a5656a15..471080e6 100644 --- a/secondorder/erroranalysisl2h1.html +++ b/secondorder/erroranalysisl2h1.html @@ -32,7 +32,7 @@ - + @@ -62,6 +62,8 @@ + + @@ -123,8 +125,6 @@ - -
    @@ -174,7 +174,7 @@ @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -406,6 +438,24 @@

    Error Analysis in \(L_2 \times H^ @@ -434,7 +484,7 @@

    Error Analysis in \(L_2 \times H^

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/secondorder/finiteelements.html b/secondorder/finiteelements.html index 03201ceb..6c9f34e2 100644 --- a/secondorder/finiteelements.html +++ b/secondorder/finiteelements.html @@ -32,7 +32,7 @@ - + @@ -59,9 +59,14 @@ + + + + + @@ -123,8 +128,6 @@ - -
    @@ -174,7 +177,7 @@ @@ -184,6 +187,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -448,6 +483,11 @@

    Piola Transformation +
    +
    BaseWebGuiScene
+
    +
    +
    @@ -459,6 +499,8 @@

    Piola Transformation

    +
    +
    @@ -473,6 +515,12 @@

    Piola Transformation

    +
    +
    dofs on edge 22: (22, 86, 87)
+dofs on face 0: (126, 127, 128)
+
    +
    +

    With RT=True we get the Raviart-Thomas space, which is \([P^k]^d + P^k x\):

    @@ -485,6 +533,11 @@

    Piola Transformation

    +
    +
    err = 4.680363989194429e-16
+
    +
    +

    @@ -519,6 +572,24 @@

    Piola Transformation
    + + +
    +

    previous

    +

    The function space \(H(\operatorname{div})\)

    +
    +     + +
    +

    next

    +

    Finite Element Error Analysis

    +
    + +
    @@ -560,7 +631,7 @@

    Piola Transformation - © Copyright 2023. + © Copyright 2024.

    diff --git a/secondorder/framework.html b/secondorder/framework.html index 48a16b6d..5b786381 100644 --- a/secondorder/framework.html +++ b/secondorder/framework.html @@ -32,7 +32,7 @@ - + @@ -59,9 +59,14 @@ + + + + + @@ -123,8 +128,6 @@ - -
    @@ -174,7 +177,7 @@ @@ -184,6 +187,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -420,6 +455,11 @@

    Dual mixed formulation +
    +
    <ngsolve.comp.Mesh at 0x118943a70>
+
    +
    +
    @@ -441,6 +481,8 @@

    Dual mixed formulation

    +
    +
    @@ -466,6 +508,8 @@

    Dual mixed formulation

    +
    +

    We observe that the normal component of the flux is continuous arcross discontinuous coefficients. As we will see, this is a property of the function space \(H(\opdiv)\).

    @@ -514,6 +558,24 @@

    Primal mixed formulation
    + + +
    +

    previous

    +

    18. H(div)-conforming Stokes

    +
    +     + +
    +

    next

    +

    The function space \(H(\operatorname{div})\)

    +
    + +
    @@ -556,7 +618,7 @@

    Primal mixed formulation - © Copyright 2023. + © Copyright 2024.

    diff --git a/secondorder/hdiv.html b/secondorder/hdiv.html index c34b347c..cffcf08e 100644 --- a/secondorder/hdiv.html +++ b/secondorder/hdiv.html @@ -32,7 +32,7 @@ - + @@ -62,6 +62,8 @@ + + @@ -123,8 +125,6 @@ - -
    @@ -174,7 +174,7 @@ @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -473,6 +505,24 @@

    \(H(\opdiv)\) on sub-domai @@ -515,7 +565,7 @@

    \(H(\opdiv)\) on sub-domai

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/secondorder/hybridization.html b/secondorder/hybridization.html index d6996085..4654580b 100644 --- a/secondorder/hybridization.html +++ b/secondorder/hybridization.html @@ -32,7 +32,7 @@ - + @@ -59,9 +59,14 @@ + + + + + @@ -123,8 +128,6 @@ - -
    @@ -174,7 +177,7 @@ @@ -184,6 +187,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -471,34 +506,34 @@

    Hybridization Techniques -
    CG iteration 1, residual = 0.3766721408060262     
-CG iteration 2, residual = 0.10907102688107831     
-CG iteration 3, residual = 0.02762373762050006     
-CG iteration 4, residual = 0.009460904839439215     
-CG iteration 5, residual = 0.0030401178721749075     
-CG iteration 6, residual = 0.0008737003721822246     
-CG iteration 7, residual = 0.00036488495834927087     
-CG iteration 8, residual = 0.00010180163060399381     
-CG iteration 9, residual = 4.454444795606642e-05     
-CG iteration 10, residual = 1.0276081830023155e-05     
-CG iteration 11, residual = 3.2066421202691073e-06     
-CG iteration 12, residual = 7.934242177194006e-07     
-CG iteration 13, residual = 2.7503719049183633e-07     
-CG iteration 14, residual = 7.87016254760295e-08     
-CG iteration 15, residual = 2.591403235842198e-08     
-CG iteration 16, residual = 7.189732121846986e-09     
-CG iteration 17, residual = 2.4887181875054425e-09     
-CG iteration 18, residual = 7.169826660237978e-10     
-CG iteration 19, residual = 2.1905224129850516e-10     
-CG iteration 20, residual = 8.461686423769498e-11     
-CG iteration 21, residual = 2.052136853205269e-11     
-CG iteration 22, residual = 6.675385439594928e-12     
-CG iteration 23, residual = 1.3534635900852246e-12     
-CG iteration 24, residual = 4.747730301285136e-13     
-CG iteration 25, residual = 8.842243746323985e-14     
+
    CG iteration 1, residual = 0.37663133100619084     
+CG iteration 2, residual = 0.10608193520813987     
+CG iteration 3, residual = 0.02537449045713504     
+CG iteration 4, residual = 0.009211818637816978     
+CG iteration 5, residual = 0.002769829833990174     
+CG iteration 6, residual = 0.0007513083315122609     
+CG iteration 7, residual = 0.0003100098687187831     
+CG iteration 8, residual = 9.544144155753213e-05     
+CG iteration 9, residual = 4.417241324804755e-05     
+CG iteration 10, residual = 9.963174584217294e-06     
+CG iteration 11, residual = 3.4749995026727215e-06     
+CG iteration 12, residual = 1.0274167917763552e-06     
+CG iteration 13, residual = 3.3480564361458973e-07     
+CG iteration 14, residual = 9.379824934894385e-08     
+CG iteration 15, residual = 3.11656085663382e-08     
+CG iteration 16, residual = 1.0690433729174157e-08     
+CG iteration 17, residual = 2.879597282713182e-09     
+CG iteration 18, residual = 8.274291627498397e-10     
+CG iteration 19, residual = 2.439466447377823e-10     
+CG iteration 20, residual = 8.727593166439224e-11     
+CG iteration 21, residual = 2.054743965009441e-11     
+CG iteration 22, residual = 6.466548278024554e-12     
+CG iteration 23, residual = 1.5745792118100376e-12     
+CG iteration 24, residual = 5.476348443116634e-13     
+CG iteration 25, residual = 1.084394996297869e-13     
 
    
 
    
-
    BaseWebGuiScene
+
    BaseWebGuiScene
 
    
 
    
 
    
@@ -535,6 +570,24 @@ 
    Hybridization Techniques
                   
 
    
+    
+      
+      
    
+        
    previous
    
+        
    Error Analysis in \(L_2 \times H^1\)
    
+      
    
+        
+    
+      
    
+        
    next
    
+        
    19. Basic Iterative Methods
    
+      
    
+      
+        
 
    
                 
               
@@ -563,7 +616,7 @@ 
    Hybridization Techniques
     
-      © Copyright 2023.
+      © Copyright 2024.
       
    
     
   
    
diff --git a/sobolev/Friedrichs.html b/sobolev/Friedrichs.html
index 356f59b2..9337a40a 100644
--- a/sobolev/Friedrichs.html
+++ b/sobolev/Friedrichs.html
@@ -32,7 +32,7 @@
     
     
     
-    
+    
     
     
   
@@ -174,7 +174,7 @@
             
@@ -184,6 +184,38 @@
 
  • 2. Boundary Conditions
    • 
 
  • 3. Variable Coefficients
    • 
 
  • 4. Iterative Solvers
    • 
+
+
    Mixed Finite Element Methods
    
+
+
    Discontinuous Galerkin Methods
    
+
+
    Iteration Methods
    
+
 
     
    
@@ -441,7 +473,7 @@ 
    Friedrichs’ Inequality
     
-      © Copyright 2023.
+      © Copyright 2024.
       
    
     
   
    
diff --git a/sobolev/Untitled.html b/sobolev/Untitled.html
index d3ae375e..d7cdfa2d 100644
--- a/sobolev/Untitled.html
+++ b/sobolev/Untitled.html
@@ -32,7 +32,7 @@
     
     
     
-    
+    
     
     
   
@@ -172,7 +172,7 @@
             
@@ -182,6 +182,38 @@
 
  • 2. Boundary Conditions
    • 
 
  • 3. Variable Coefficients
    • 
 
  • 4. Iterative Solvers
    • 
+
+
    Mixed Finite Element Methods
    
+
+
    Discontinuous Galerkin Methods
    
+
+
    Iteration Methods
    
+
    @@ -414,7 +446,7 @@

    Contents

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/sobolev/Untitled1.html b/sobolev/Untitled1.html index e24be513..0b702fdc 100644 --- a/sobolev/Untitled1.html +++ b/sobolev/Untitled1.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@ @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    +
    @@ -414,7 +446,7 @@

    Contents

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/sobolev/Untitled2.html b/sobolev/Untitled2.html index 4b613bef..811b16e4 100644 --- a/sobolev/Untitled2.html +++ b/sobolev/Untitled2.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@ @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -402,7 +434,7 @@

    Friedrichs’ Inequality - © Copyright 2023. + © Copyright 2024.

    diff --git a/sobolev/poincare.html b/sobolev/poincare.html index 8d34921b..f8091149 100644 --- a/sobolev/poincare.html +++ b/sobolev/poincare.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -438,7 +470,7 @@

    The Poincaré inequality - © Copyright 2023. + © Copyright 2024.

    diff --git a/sobolev/trace.html b/sobolev/trace.html index 031248d7..953c546b 100644 --- a/sobolev/trace.html +++ b/sobolev/trace.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -533,7 +565,7 @@

    Proof of the trace inequality: - © Copyright 2023. + © Copyright 2024.

    diff --git a/subspacecorrection/Untitled.html b/subspacecorrection/Untitled.html index dd3857ef..040210a2 100644 --- a/subspacecorrection/Untitled.html +++ b/subspacecorrection/Untitled.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@     @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -414,7 +446,7 @@

    Contents

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/subspacecorrection/asm.html b/subspacecorrection/asm.html index fca0a1ff..8de7b086 100644 --- a/subspacecorrection/asm.html +++ b/subspacecorrection/asm.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@ @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -689,7 +721,7 @@

    The upper-bound by the overlap - © Copyright 2023. + © Copyright 2024.

    diff --git a/subspacecorrection/examples.html b/subspacecorrection/examples.html index 9b11b9a8..dcab8a6f 100644 --- a/subspacecorrection/examples.html +++ b/subspacecorrection/examples.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -682,7 +714,7 @@

    \(H^1\)-norm with small - © Copyright 2023. + © Copyright 2024.

    diff --git a/subspacecorrection/exercise20210324.html b/subspacecorrection/exercise20210324.html index 40ba5cc1..d41f03f1 100644 --- a/subspacecorrection/exercise20210324.html +++ b/subspacecorrection/exercise20210324.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@ @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -456,7 +488,7 @@

    Team B:# - © Copyright 2023. + © Copyright 2024.

    diff --git a/subspacecorrection/minimaldd.html b/subspacecorrection/minimaldd.html index 99724e51..7fb3e919 100644 --- a/subspacecorrection/minimaldd.html +++ b/subspacecorrection/minimaldd.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -773,7 +805,7 @@

    Graph-based mesh partitioning - © Copyright 2023. + © Copyright 2024.

    diff --git a/subspacecorrection/overlapping.html b/subspacecorrection/overlapping.html index 2e6794a2..c415b421 100644 --- a/subspacecorrection/overlapping.html +++ b/subspacecorrection/overlapping.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -773,7 +805,7 @@

    Comparison to DD with minimal overlap - © Copyright 2023. + © Copyright 2024.

    diff --git a/timedependent/ODEs/runge_kutta.html b/timedependent/ODEs/runge_kutta.html index 664203a4..bb836b38 100644 --- a/timedependent/ODEs/runge_kutta.html +++ b/timedependent/ODEs/runge_kutta.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -612,7 +644,7 @@

    Exercise: - © Copyright 2023. + © Copyright 2024.

    diff --git a/timedependent/ODEs/singlestep.html b/timedependent/ODEs/singlestep.html index b65cac2f..842d4c06 100644 --- a/timedependent/ODEs/singlestep.html +++ b/timedependent/ODEs/singlestep.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -763,7 +795,7 @@

    Exercises: - © Copyright 2023. + © Copyright 2024.

    diff --git a/timedependent/intro/Untitled.html b/timedependent/intro/Untitled.html index 9a7ca6ec..ded3fbdd 100644 --- a/timedependent/intro/Untitled.html +++ b/timedependent/intro/Untitled.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@     @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -437,7 +469,7 @@

    Contents

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/timedependent/intro/heatequation.html b/timedependent/intro/heatequation.html index aacb82c9..4dc9bd72 100644 --- a/timedependent/intro/heatequation.html +++ b/timedependent/intro/heatequation.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@ @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -576,7 +608,7 @@

    Implicit Euler time-stepping - © Copyright 2023. + © Copyright 2024.

    diff --git a/timedependent/intro/waveequation.html b/timedependent/intro/waveequation.html index a71ffbfc..598e960d 100644 --- a/timedependent/intro/waveequation.html +++ b/timedependent/intro/waveequation.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -522,7 +554,7 @@

    Newmark time-stepping method - © Copyright 2023. + © Copyright 2024.

    diff --git a/timedependent/parabolic/exponential.html b/timedependent/parabolic/exponential.html index b5d1d2ee..f26e34fd 100644 --- a/timedependent/parabolic/exponential.html +++ b/timedependent/parabolic/exponential.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@     @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -563,7 +595,7 @@

    Exponential Integrators for Parabolic Equations - © Copyright 2023. + © Copyright 2024.

    diff --git a/timedependent/waves/Maxwell-DG.html b/timedependent/waves/Maxwell-DG.html index 80d22d9b..5a0c508a 100644 --- a/timedependent/waves/Maxwell-DG.html +++ b/timedependent/waves/Maxwell-DG.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@     @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -508,7 +540,7 @@

    Contents

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/timedependent/waves/Untitled.html b/timedependent/waves/Untitled.html index 3dbf0faa..a5434996 100644 --- a/timedependent/waves/Untitled.html +++ b/timedependent/waves/Untitled.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@ @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -421,7 +453,7 @@

    Contents

    - © Copyright 2023. + © Copyright 2024.

    diff --git a/timedependent/waves/lts.html b/timedependent/waves/lts.html index 8b76383a..02effa33 100644 --- a/timedependent/waves/lts.html +++ b/timedependent/waves/lts.html @@ -32,7 +32,7 @@ - + @@ -174,7 +174,7 @@ @@ -184,6 +184,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -618,7 +650,7 @@

    Geometry with local details: - © Copyright 2023. + © Copyright 2024.

    diff --git a/timedependent/waves/ringresonator.html b/timedependent/waves/ringresonator.html index 5c17ea23..04f66f4e 100644 --- a/timedependent/waves/ringresonator.html +++ b/timedependent/waves/ringresonator.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@     @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -465,7 +497,7 @@

    Nano-optics: A ring-resonator - © Copyright 2023. + © Copyright 2024.

    diff --git a/timedependent/waves/testpml/ringresonator.html b/timedependent/waves/testpml/ringresonator.html index 4fd993b7..5c3002c2 100644 --- a/timedependent/waves/testpml/ringresonator.html +++ b/timedependent/waves/testpml/ringresonator.html @@ -32,7 +32,7 @@ - + @@ -172,7 +172,7 @@     @@ -182,6 +182,38 @@
  • 2. Boundary Conditions
  • 3. Variable Coefficients
  • 4. Iterative Solvers
    • + +

    Mixed Finite Element Methods

    + +

    Discontinuous Galerkin Methods

    + +

    Iteration Methods

    + @@ -350,7 +382,7 @@

    Nano-optics: A ring-resonator - +
    @@ -385,7 +417,7 @@

    Nano-optics: A ring-resonator -
    ---------------------------------------------------------------------------
+
    ---------------------------------------------------------------------------
 NameError                                 Traceback (most recent call last)
 Cell In[3], line 9
       7 while t < tend:
@@ -469,7 +501,7 @@ 
    Nano-optics: A ring-resonator
     
-      © Copyright 2023.
+      © Copyright 2024.
       
    
     
   
    
diff --git a/timedependent/waves/wave-leapfrogDG.html b/timedependent/waves/wave-leapfrogDG.html
index 25f1c0d0..a897dc4f 100644
--- a/timedependent/waves/wave-leapfrogDG.html
+++ b/timedependent/waves/wave-leapfrogDG.html
@@ -32,7 +32,7 @@
     
     
     
-    
+    
     
     
   
@@ -174,7 +174,7 @@
             
@@ -184,6 +184,38 @@
 
  • 2. Boundary Conditions
    • 
 
  • 3. Variable Coefficients
    • 
 
  • 4. Iterative Solvers
    • 
+
+
    Mixed Finite Element Methods
    
+
+
    Discontinuous Galerkin Methods
    
+
+
    Iteration Methods
    
+
 
     
    
@@ -823,7 +855,7 @@ 
    Exercise:
     
-      © Copyright 2023.
+      © Copyright 2024.