From 7f4ad95f870bb050588340231d5246256f1fc0dc Mon Sep 17 00:00:00 2001 From: Joachim Schoeberl Date: Mon, 11 Mar 2024 07:36:15 +0100 Subject: [PATCH] Update documentation --- _sources/abstracttheory/infsup.ipynb | 4 ++-- abstracttheory/infsup.html | 10 +++++----- searchindex.js | 2 +- 3 files changed, 8 insertions(+), 8 deletions(-) diff --git a/_sources/abstracttheory/infsup.ipynb b/_sources/abstracttheory/infsup.ipynb index 2607d893..0a284443 100644 --- a/_sources/abstracttheory/infsup.ipynb +++ b/_sources/abstracttheory/infsup.ipynb @@ -39,7 +39,7 @@ "\\| B u \\|_{W^\\ast} \\geq \\beta_1 \\| u \\|_V.\n", "$$\n", "\n", - "We immediately obtain that $B$ is one to one, since\n", + "We immediately obtain that $B$ is one to one (aka injective), since\n", "\n", "$$\n", "B u = 0 \\Rightarrow u = 0\n", @@ -52,7 +52,7 @@ "converges to some $u \\in V$. By continuity of $B$, the sequence $B u^n$ converges to $B u \\in W^\\ast$.\n", " $\\Box$\n", "\n", - "The inf-sup condition does not imply that $B$ is onto $W^\\ast$. To insure that, we can pose an inf-sup condition the other way around:\n", + "The inf-sup condition does not imply that $B$ is onto $W^\\ast$. To insure that, we can pose an inf-sup condition the other way around (aka surjective):\n", "\n", "$$\n", "\\inf_{v \\in W \\atop v \\neq 0} \\sup_{u \\in V \\atop u \\neq 0}\n", diff --git a/abstracttheory/infsup.html b/abstracttheory/infsup.html index 7ddc5fa6..2bcc498b 100644 --- a/abstracttheory/infsup.html +++ b/abstracttheory/infsup.html @@ -32,7 +32,7 @@ - + @@ -507,7 +507,7 @@

11. Inf-sup stable variational problems< \[ \| B u \|_{W^\ast} \geq \beta_1 \| u \|_V. \] -

We immediately obtain that \(B\) is one to one, since

+

We immediately obtain that \(B\) is one to one (aka injective), since

\[ B u = 0 \Rightarrow u = 0 @@ -519,7 +519,7 @@

11. Inf-sup stable variational problems<

Proof: Let \(B u^n\) be a Cauchy sequence in \(W^\ast\). From \(\| Bu \| \geq \beta_1 \| u \|\) we conclude that also \(u^n\) is Cauchy in \(V\). Since \(V\) is complete, \(u_n\) converges to some \(u \in V\). By continuity of \(B\), the sequence \(B u^n\) converges to \(B u \in W^\ast\). \(\Box\)

-

The inf-sup condition does not imply that \(B\) is onto \(W^\ast\). To insure that, we can pose an inf-sup condition the other way around:

+

The inf-sup condition does not imply that \(B\) is onto \(W^\ast\). To insure that, we can pose an inf-sup condition the other way around (aka surjective):

\[ \inf_{v \in W \atop v \neq 0} \sup_{u \in V \atop u \neq 0} @@ -636,8 +636,8 @@

11.1. Approximation of inf-sup stable va of the Babuška-Aziz theorem. Furthermore, \(B(\cdot,\cdot)\) fulfills the discrete inf-sup condition with bound \(\beta_{1h}\). Then there holds the quasi-optimal error estimate

-
-()#\[\begin{equation} +
+()#\[\begin{equation} \| u - u_h \| \leq (1 + \beta_2 / \beta_{1h}) \inf_{v_h \in V_h} \| u - v_h \| \end{equation}\]
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6, 11, 22, 23, 36, 38, 50, 51, 53, 54, 55, 56, 58, 59, 60, 65, 66, 67, 71, 74, 78, 79, 81, 83, 86, 88, 95, 97, 99, 104, 105, 114], "cc": [5, 23, 36, 59, 71, 79, 81, 88, 104, 114], "end": [5, 6, 8, 9, 11, 12, 15, 23, 26, 27, 29, 31, 32, 33, 34, 35, 36, 38, 39, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 71, 76, 77, 78, 79, 80, 81, 83, 84, 86, 87, 88, 89, 94, 95, 96, 97, 98, 99, 100, 103, 104, 105, 106, 107, 108, 111, 114], "c": [5, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 32, 35, 36, 40, 44, 45, 46, 47, 49, 51, 52, 54, 61, 62, 63, 65, 71, 75, 77, 78, 79, 80, 81, 82, 87, 88, 89, 92, 93, 94, 95, 96, 97, 98, 99, 100, 102, 103, 104, 105, 106], "diff": 5, "conv": [5, 16], "ep": [5, 80, 100], "adiff": 5, "aconv": 5, "mstar": [5, 106, 107, 108], "asvector": [5, 40], "0x10dcfe070": 5, "convu": [5, 16], "keyboardinterrupt": 5, "traceback": [5, 113], "most": [5, 45, 96, 102, 104, 113], "recent": [5, 44, 60, 111, 113], 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102, 103], "follow": [6, 8, 10, 11, 29, 31, 32, 33, 34, 35, 38, 39, 44, 45, 49, 51, 52, 61, 62, 63, 64, 71, 77, 80, 84, 85, 88, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 114], "wise": [6, 14, 23, 32, 35, 56, 58, 68, 74, 83, 85, 100, 114], "smooth": [6, 8, 9, 12, 31, 39, 58, 61, 62, 87, 95, 96, 97, 100], "evei": 6, "sum_": [6, 8, 9, 11, 12, 23, 31, 32, 33, 34, 36, 38, 39, 40, 45, 47, 49, 60, 63, 64, 74, 77, 84, 87, 88, 96, 97, 98, 99, 100, 104, 106, 114], "subset": [6, 8, 9, 11, 12, 14, 25, 27, 29, 32, 34, 35, 38, 39, 45, 47, 52, 61, 62, 64, 65, 66, 74, 77, 80, 81, 85, 89, 92, 94, 95, 97, 98, 99, 100, 103, 104, 105, 106], "b_n": 6, "outward": 6, "decis": 6, "blow": 6, "up": [6, 8, 11, 13, 14, 23, 26, 29, 47, 49, 56, 61, 64, 77, 80, 94, 96, 98, 103, 104, 105, 106, 114], "cl": [6, 11, 34, 95], "u_t": [6, 60], "outflow": [6, 15, 17, 18, 55], "v_h": [6, 8, 11, 12, 23, 26, 27, 29, 32, 33, 34, 52, 60, 63, 66, 74, 80, 81, 83, 84, 88, 99, 100, 103, 105, 106], "u_h": [6, 8, 11, 12, 26, 27, 29, 31, 32, 33, 34, 52, 60, 63, 74, 80, 83, 84, 88, 99, 103, 105, 106], "all": [6, 8, 9, 10, 14, 19, 23, 25, 27, 28, 29, 30, 31, 32, 34, 35, 37, 44, 47, 49, 50, 51, 52, 61, 63, 64, 67, 70, 71, 72, 74, 80, 87, 96, 97, 98, 99, 100, 102, 103, 104, 105, 106, 108, 114], "6": [6, 13, 20, 22, 36, 37, 38, 42, 45, 46, 53, 61, 68, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 85, 88, 102, 104, 105, 108, 111], "28": [6, 20, 36, 37, 46, 74, 75, 78, 80, 81, 82], "hybrid": [7, 43, 44, 52, 65, 70], "lehrenfeld": [7, 70], "sch\u00f6berl": [7, 43, 44, 60, 69, 70, 80, 112, 113], "2016": 7, "bdm": [7, 32, 85], "k": [7, 8, 25, 26, 32, 39, 45, 46, 47, 48, 49, 56, 63, 64, 65, 77, 81, 83, 84, 85, 88, 95, 96, 97, 98, 105], "k_": 7, "splinegeometri": [7, 42, 60, 78, 82, 86], "geo": [7, 35, 36, 37, 42, 60, 68, 72, 76, 78, 79, 80, 82, 86, 102, 111, 114], "addrectangl": [7, 42, 60, 78, 82, 86], "41": [7, 14, 15, 17, 18, 20, 21, 36, 37, 46, 74, 75, 78, 80, 81, 82], "wall": [7, 15, 17, 18, 21, 78, 82], "outlet": [7, 15, 17, 18, 21, 78, 82], "inlet": [7, 15, 17, 18, 21, 78, 82], "addcircl": [7, 42, 78, 82, 86], "leftdomain": [7, 42, 78, 82, 86], "rightdomain": [7, 42, 78, 82, 86], "cyl": [7, 15, 21, 68, 72, 75, 78, 82], "08": [7, 20, 36, 37, 46, 61, 75, 78, 80, 81, 82, 88, 108], "0x10eaa3e90": 7, "vt": 7, "vf": 7, "tangentialfacetfespac": [7, 16, 17, 18, 109], "q": [7, 9, 21, 25, 32, 51, 52, 53, 54, 56, 60, 76, 77, 78, 81, 82, 97, 98, 114], "nu": [7, 15, 17, 18, 72, 80], "tang": [7, 16, 17, 18, 60, 65, 67, 68], "thesi": [7, 57, 60, 70], "christoph": [7, 70], "page": [7, 45, 47], "71": [7, 36, 37, 46, 75, 80, 82], "invstok": 7, "uin": [7, 15, 17, 18, 21, 78, 82], "re": [7, 32, 34, 46, 48, 49, 61, 79, 106], "vel": 7, "veloc": [7, 15, 56, 57, 81, 107], "pressur": [7, 56, 80, 81], "let": [8, 11, 12, 19, 24, 25, 26, 27, 28, 29, 30, 31, 34, 35, 38, 39, 45, 46, 47, 48, 60, 61, 62, 63, 64, 74, 77, 79, 80, 81, 83, 89, 92, 94, 95, 96, 97, 98, 99, 100, 102, 103, 106], "variat": [8, 28, 43, 44, 51, 58, 65, 71, 73, 74, 76, 79, 80, 81, 83, 84, 88, 98, 99, 107], "problem": [8, 10, 11, 12, 20, 32, 33, 35, 36, 37, 38, 39, 45, 48, 49, 58, 61, 62, 63, 65, 67, 68, 73, 74, 75, 77, 79, 83, 86, 88, 94, 97, 98, 99, 100, 101, 102, 103, 105], "sub": [8, 11, 22, 23, 25, 28, 30, 32, 35, 36, 37, 38, 40, 47, 51, 52, 61, 63, 71, 74, 75, 77, 79, 80, 81, 85, 95, 100, 102, 103, 104, 105, 106], "cea": [8, 11, 62], "": [8, 9, 11, 17, 18, 24, 25, 26, 27, 28, 30, 32, 37, 43, 44, 51, 52, 54, 62, 68, 69, 70, 72, 73, 74, 75, 79, 81, 89, 97, 98, 99, 104, 105, 108, 111, 114], "lemma": [8, 9, 25, 26, 29, 30, 34, 35, 43, 61, 62, 63, 77, 80, 97, 98, 102, 103, 105], "bound": [8, 9, 11, 25, 26, 28, 29, 31, 32, 33, 34, 45, 46, 47, 49, 63, 80, 81, 94, 95, 96, 97, 98, 100, 102, 103], "best": [8, 26, 52, 65, 77], "_v": [8, 11, 25, 26, 27, 28, 29, 31, 33, 34, 51, 52, 73, 97, 98], "leq": [8, 9, 11, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 45, 46, 47, 48, 49, 51, 52, 59, 62, 63, 64, 65, 69, 77, 80, 81, 83, 87, 89, 92, 93, 94, 95, 96, 97, 98, 99, 100, 102, 103, 105, 108], "inf_": [8, 11, 26, 28, 29, 30, 35, 39, 46, 52, 60, 63, 65, 77, 80, 97, 98, 99, 100, 102, 103], "constant": [8, 9, 10, 11, 14, 26, 27, 28, 31, 32, 34, 38, 40, 49, 52, 56, 57, 61, 62, 64, 65, 74, 76, 77, 85, 89, 95, 97, 98, 100, 102, 103, 104, 105, 114], "factor": [8, 26, 29, 46, 61, 62, 64, 68, 74, 77, 80, 81, 94, 97, 98, 102, 103, 114], "ratio": 8, "take": [8, 13, 24, 27, 29, 30, 31, 34, 38, 39, 44, 52, 60, 63, 74, 76, 81, 84, 86, 96, 97, 100, 102, 105], "interpol": [8, 9, 11, 16, 34, 44, 52, 63, 64, 71, 72, 73, 74, 80, 83, 97, 103, 111, 114], "i_": [8, 9, 99], "cal": [8, 9, 11, 12, 31, 33, 34, 95, 96], "dimension": [8, 10, 25, 26, 27, 28, 29, 47, 50, 74, 77, 81, 84, 97, 98, 100, 108], "domain": [8, 9, 10, 11, 23, 31, 35, 36, 37, 40, 41, 51, 61, 63, 64, 71, 74, 79, 80, 94, 95, 96, 98, 106, 112, 113], "relat": [8, 12, 27, 45, 48, 62, 76, 98], "affin": [8, 9, 58], "transform": [8, 9, 10, 11, 43, 44, 49, 74, 79, 82, 94, 97, 100], "f_t": [8, 9, 11, 12, 23, 32, 60], "rightarrow": [8, 9, 11, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 39, 48, 51, 52, 54, 56, 64, 66, 67, 72, 73, 74, 76, 77, 80, 94, 96, 97, 98, 99, 100, 103, 105, 106, 108], "mathbb": [8, 9, 11, 12, 25, 26, 28, 29, 30, 33, 34, 35, 46, 47, 48, 49, 50, 61, 64, 72, 73, 74, 76, 79, 81, 89, 92, 94, 95, 96, 97, 98, 100, 104, 105, 108, 114], "regular": [8, 9, 10, 31, 32, 37, 51, 60, 62, 63, 74, 80, 81, 88, 95, 99, 103], "Then": [8, 11, 12, 14, 19, 23, 25, 26, 27, 28, 29, 30, 32, 34, 36, 37, 39, 44, 45, 46, 48, 51, 52, 61, 62, 63, 64, 77, 80, 81, 89, 94, 95, 96, 97, 98, 99, 100, 102, 105, 111], "hold": [8, 9, 11, 25, 26, 27, 29, 30, 31, 32, 39, 47, 48, 49, 51, 61, 62, 63, 64, 72, 74, 77, 80, 87, 94, 95, 96, 97, 98, 99, 100, 103, 105], "det": [8, 85, 97, 100, 105, 114], "align": [8, 9, 12, 26, 27, 29, 45, 46, 48, 49, 50, 77, 97, 98], "x_": [8, 11, 47], "i_m": 8, "ldot": [8, 9, 12, 28, 32, 45, 46, 47, 48, 49, 61, 64, 74, 77, 87, 95, 97, 99, 104, 105, 106, 108], "i_1": [8, 39], "j_m": 8, "j_1": 8, "b_": [8, 51, 104], "equat": [8, 11, 12, 26, 29, 31, 32, 33, 34, 36, 38, 39, 46, 50, 52, 53, 59, 63, 71, 72, 75, 76, 77, 79, 81, 86, 88, 97, 98, 99, 104, 111, 112, 113], "preceq": [8, 11, 33, 34, 62, 63, 64, 77, 80, 98, 100, 102], "proof": [8, 11, 25, 26, 27, 29, 30, 32, 34, 35, 52, 62, 63, 64, 73, 77, 80, 81, 84, 85, 87, 89, 96, 97, 98, 99, 100, 102, 103, 105], "chain": [8, 85, 97, 100], "box": [8, 11, 21, 25, 26, 27, 29, 30, 63, 72, 75, 76, 77, 80, 82, 94, 96, 97, 98, 102, 114], "diamet": 8, "h_t": [8, 11, 32, 34], "operatornam": [8, 23, 25, 26, 27, 29, 36, 39, 44, 47, 51, 52, 54, 55, 56, 57, 59, 60, 61, 64, 72, 74, 79, 80, 81, 83, 94, 95, 97, 98, 99, 100, 102, 103, 105, 114], "diam": [8, 103], "triangul": [8, 9, 74], "shape": [8, 9, 10, 15, 17, 18, 21, 23, 31, 32, 40, 44, 61, 74, 75, 76, 82, 102, 103, 111, 114], "fulfil": [8, 26, 27, 29, 31, 34, 97], "succeq": 8, "sim": [8, 71], "If": [8, 9, 10, 11, 19, 23, 24, 25, 26, 27, 30, 31, 34, 39, 44, 45, 46, 47, 49, 51, 52, 61, 62, 64, 67, 74, 77, 80, 81, 88, 94, 95, 97, 98, 99, 103, 104, 105, 106, 114], "studi": [8, 38, 56, 73, 97], "converg": [8, 25, 26, 29, 30, 31, 36, 37, 39, 46, 48, 49, 54, 63, 73, 75, 77, 80, 96, 97, 98, 108], "famili": [8, 85], "decreas": [8, 38, 46, 49], "case": [8, 9, 12, 22, 26, 27, 28, 29, 33, 34, 49, 52, 56, 58, 63, 65, 67, 71, 74, 76, 80, 81, 96, 97, 98, 99, 100, 108], "geq": [8, 11, 25, 26, 28, 29, 30, 31, 32, 45, 49, 51, 52, 62, 64, 65, 77, 81, 84, 97], "refer": [8, 9, 10, 11, 34, 50, 60, 62, 63, 85, 100, 105, 114], "triangl": [8, 9, 31, 32, 34, 52, 64, 80, 83, 100, 111], "b_t": 8, "simeq": [8, 11, 34, 97, 98], "main": [8, 32], "applic": [8, 11, 31, 33, 43, 44, 47, 61, 70, 78, 79, 114], "brambl": [8, 11, 34, 43, 44, 62, 77, 82, 98], "hilbert": [8, 11, 25, 26, 27, 28, 29, 30, 34, 51, 52, 54, 74, 77, 80, 96, 97, 98, 106], "sometim": 8, "itself": [8, 14, 19, 99, 107], "v_t": [8, 9, 23, 32, 60, 85], "psi_t": [8, 9], "contain": [8, 9, 22, 25, 31, 32, 34, 39, 47, 74, 80, 95, 96, 103, 105], "i_t": [8, 9], "first": [8, 10, 12, 24, 25, 27, 30, 31, 34, 36, 44, 45, 46, 49, 51, 52, 53, 54, 55, 56, 59, 60, 62, 63, 68, 74, 77, 79, 80, 81, 86, 88, 94, 95, 97, 98, 100, 103, 105, 107, 108, 111, 114], "prove": [8, 11, 27, 28, 30, 32, 46, 49, 51, 62, 63, 77, 80, 81, 94, 96, 97, 98, 99, 100, 103], "psi_": [8, 9, 62], "varphi_": [8, 62, 64, 77, 87, 100], "project": [8, 9, 26, 27, 35, 43, 44, 62, 63, 80, 83, 102, 103, 106, 111], "l": [8, 11, 13, 14, 17, 18, 22, 25, 26, 27, 29, 44, 45, 49, 51, 61, 62, 63, 64, 66, 75, 77, 80, 81, 83, 86, 96, 97, 98, 99, 103, 104, 105, 111], "result": [8, 11, 12, 14, 19, 23, 35, 49, 60, 63, 64, 67, 69, 74, 77, 80, 94, 96, 97, 98, 100, 103], "theorem": [8, 11, 26, 28, 29, 30, 34, 35, 43, 44, 51, 52, 62, 63, 64, 74, 77, 80, 81, 87, 96, 98, 99, 103], "v_": [8, 9, 11, 12, 29, 32, 40, 51, 61, 62, 65, 77, 80, 97], "local": [8, 9, 10, 12, 14, 20, 22, 23, 31, 32, 33, 34, 35, 38, 40, 43, 44, 57, 61, 62, 64, 74, 75, 77, 78, 80, 95, 103], "each": [8, 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[[19, "introduction-to-mpi-with-mpi4py"]], "The MPI library": [[19, "the-mpi-library"]], "Iteration methods in parallel": [[20, "iteration-methods-in-parallel"]], "Richardson iteration": [[20, "richardson-iteration"]], "Solving Stokes in parallel": [[21, "solving-stokes-in-parallel"]], "Using PETSc": [[22, "using-petsc"]], "Consistent and Distributed Vectors": [[23, "consistent-and-distributed-vectors"]], "Consistent vectors:": [[23, "consistent-vectors"]], "Distributed vectors and matrices": [[23, "distributed-vectors-and-matrices"]], "Inner products:": [[23, "inner-products"]], "Matrix vector multiplication:": [[23, "matrix-vector-multiplication"]], "Vector operations:": [[23, "vector-operations"]], "This repository contains an interactive introduction to the Finite Element Method": [[24, "this-repository-contains-an-interactive-introduction-to-the-finite-element-method"]], "Riesz representation theorem and symmetric variational problems": [[27, "riesz-representation-theorem-and-symmetric-variational-problems"]], "Symmetric variational problems": [[27, "symmetric-variational-problems"]], "A posteriori error estimates": [[31, "a-posteriori-error-estimates"], [44, null]], "The Zienkiewicz Zhu error estimator": [[31, "the-zienkiewicz-zhu-error-estimator"]], "Equilibrated Residual Error Estimates": [[32, "equilibrated-residual-error-estimates"]], "Goal driven error estimates": [[33, "goal-driven-error-estimates"]], "The residual error estimator": [[34, "the-residual-error-estimator"]], "BDDC - Preconditioner": [[35, "bddc-preconditioner"]], "FETI methods": [[36, "feti-methods"]], "Preconditioner for S": [[36, "preconditioner-for-s"]], "FETI-DP": [[37, "feti-dp"]], "Introduction to Non-overlapping Domain Decomposition": [[38, "introduction-to-non-overlapping-domain-decomposition"]], "Domain Decomposition with Lagrange parameters": [[38, "domain-decomposition-with-lagrange-parameters"]], "Traces spaces": [[39, "traces-spaces"]], "Natural trace space": [[39, "natural-trace-space"]], "Interpolation space H^s": [[39, "interpolation-space-h-s"]], "Trace norm on bottom edge": [[39, "trace-norm-on-bottom-edge"]], "Trace norm on boundary sub-domains": [[39, "trace-norm-on-boundary-sub-domains"]], "Implement a parallel BDDC preconditioner": [[40, "implement-a-parallel-bddc-preconditioner"]], "Helmholtz Equation": [[41, "helmholtz-equation"]], "Grating": [[42, "grating"]], "An Interactive Introduction to the Finite Element Method": [[43, "an-interactive-introduction-to-the-finite-element-method"], [44, "an-interactive-introduction-to-the-finite-element-method"]], "The Galerkin Method": [[43, "the-galerkin-method"], [44, null]], "Abstract Theory": [[43, "abstract-theory"], [44, null], [51, "abstract-theory"]], "Mixed Finite Element Methods": [[43, "mixed-finite-element-methods"], [44, null]], "Discontinuous Galerkin Methods": [[43, "discontinuous-galerkin-methods"], [44, null]], "Mixed Methods for Second Order Equations": [[43, "mixed-methods-for-second-order-equations"], [44, null]], "Mixed Methods for Elasticity": [[43, "mixed-methods-for-elasticity"]], "Mixed Methods for Plates and Shells": [[43, "mixed-methods-for-plates-and-shells"]], "The Helmholtz equation": [[43, "the-helmholtz-equation"]], "Iterative Solvers": [[43, "iterative-solvers"], [75, "iterative-solvers"]], "Iteration Methods": [[43, "iteration-methods"], [44, null]], "Sub-space Correction Methods": [[43, "sub-space-correction-methods"], [44, null]], "Multigrid Methods": [[43, "multigrid-methods"], [44, null]], "Saddle-point Problems": [[43, "saddle-point-problems"], [44, null]], "Non-overlapping Domain Decomposition Methods": [[43, "non-overlapping-domain-decomposition-methods"], [44, null]], "Parallel Solvers": [[43, "parallel-solvers"], [44, null]], "Time-dependent Problems": [[43, "time-dependent-problems"]], "A practical introduction": [[43, "a-practical-introduction"]], "Ordinary differential equations": [[43, "ordinary-differential-equations"]], "Numerical analysis of Parabolic Equations": [[43, "numerical-analysis-of-parabolic-equations"]], "Numerical analysis of Wave Equations": [[43, "numerical-analysis-of-wave-equations"]], "Additional methods": [[43, "additional-methods"]], "Appendix": [[43, "appendix"]], "Some inequalities in Sobolev spaces": [[43, "some-inequalities-in-sobolev-spaces"]], "Literature": [[44, "literature"]], "Installing NGSolve": [[44, "installing-ngsolve"]], "Sobolev Spaces": [[44, null]], "High Order Finite Elements": [[44, null]], "The Chebyshev Method": [[45, "the-chebyshev-method"]], "Chebyshev polynomials": [[45, "chebyshev-polynomials"]], "The Chebyshev iteration": [[45, "the-chebyshev-iteration"]], "The Richardson Iteration": [[46, "the-richardson-iteration"]], "Optimizing the relaxation parameter \\alpha": [[46, "optimizing-the-relaxation-parameter-alpha"]], "Experiments with the Richardson iteration": [[46, "experiments-with-the-richardson-iteration"]], "Conjugate Gradients": [[47, "conjugate-gradients"]], "Solving the minimization problem": [[47, "solving-the-minimization-problem"]], "Expanding the Krylov-space": [[47, "expanding-the-krylov-space"]], "The Gradient Method": [[48, "the-gradient-method"]], "Preconditioning": [[49, "preconditioning"]], "The preconditioned Richardson iteration": [[49, "the-preconditioned-richardson-iteration"]], "The preconditioned gradient method": [[49, "the-preconditioned-gradient-method"]], "Jacobi and Gauss Seidel Preconditioners": [[49, "jacobi-and-gauss-seidel-preconditioners"]], "Basic Iterative Methods": [[50, "basic-iterative-methods"]], "Constrained minimization problem": [[51, "constrained-minimization-problem"]], "Stokes equation within the abstract theory": [[51, "stokes-equation-within-the-abstract-theory"]], "Dirichlet boundary conditions as mixed system": [[51, "dirichlet-boundary-conditions-as-mixed-system"]], "Mixed method for second order equation": [[51, "mixed-method-for-second-order-equation"]], "Abstract theory for mixed finite element methods": [[52, "abstract-theory-for-mixed-finite-element-methods"]], "Finite element error estimates": [[52, "finite-element-error-estimates"]], "Proving discrete LBB by the Fortin operator": [[52, "proving-discrete-lbb-by-the-fortin-operator"]], "Example: Finite elements for Stokes": [[52, "example-finite-elements-for-stokes"]], "Boundary Conditions": [[53, "boundary-conditions"], [71, "boundary-conditions"]], "Parameter Dependent Problems": [[54, "parameter-dependent-problems"], [80, "parameter-dependent-problems"]], "Example: Dirichlet boundary condition by penalty": [[54, "example-dirichlet-boundary-condition-by-penalty"]], "Example: Nearly incompressible materials": [[54, "example-nearly-incompressible-materials"]], "Mixed Methods for second order equations": [[55, "mixed-methods-for-second-order-equations"]], "Stokes Equation": [[56, "stokes-equation"]], "Variational Formulation": [[56, "variational-formulation"]], "Finite Element Spaces": [[56, "finite-element-spaces"]], "\\DeclareMathOperator{\\opdiv}{div}\nHellinger Reissner mixed formulation": [[58, "declaremathoperator-opdiv-div-hellinger-reissner-mixed-formulation"]], "Primal mixed method:": [[58, "primal-mixed-method"]], "Dual mixed method": [[58, "dual-mixed-method"]], "\\DeclareMathOperator{\\opdiv}{div}\n\\DeclareMathOperator{\\opcurl}{curl}\n\\DeclareMathOperator{\\eps}{\\varepsilon}\nReduced symmetry methods": [[59, "declaremathoperator-opdiv-div-declaremathoperator-opcurl-curl-declaremathoperator-eps-varepsilon-reduced-symmetry-methods"]], "Tangential displacement normal normal stress continuous finite elements": [[60, "tangential-displacement-normal-normal-stress-continuous-finite-elements"]], "The divergence of nn-continuous piece-wise smooth functions:": [[60, "the-divergence-of-nn-continuous-piece-wise-smooth-functions"]], "TDNNS Variational formulation:": [[60, "tdnns-variational-formulation"]], "Error estimates:": [[60, "error-estimates"]], "Multigrid and Multilevel Methods": [[61, "multigrid-and-multilevel-methods"]], "Multilevel preconditioner": [[61, "multilevel-preconditioner"]], "Multigrid Preconditioning": [[61, "multigrid-preconditioning"]], "Projection matrices from the finest level": [[61, "projection-matrices-from-the-finest-level"]], "Analysis of the Multigrid Iteration": [[62, "analysis-of-the-multigrid-iteration"]], "The Algorithm": [[62, "the-algorithm"]], "The Approximation Property": [[62, "the-approximation-property"]], "The Smothing Property": [[62, "the-smothing-property"]], "Optimal convergence of the V-cycle": [[62, "optimal-convergence-of-the-v-cycle"]], "Analysis of the multi-level preconditioner": [[63, "analysis-of-the-multi-level-preconditioner"]], "Nearly optimal analysis of the ML - preconditioner": [[63, "nearly-optimal-analysis-of-the-ml-preconditioner"]], "Optimal analysis of the multi-level preconditioner": [[63, "optimal-analysis-of-the-multi-level-preconditioner"]], "Multi-level Extension": [[64, "multi-level-extension"]], "Efficiently computable multi-level decomposition": [[64, "efficiently-computable-multi-level-decomposition"]], "Algorithm": [[64, "algorithm"]], "Extending boundary data": [[64, "extending-boundary-data"]], "The Hellan Herrmann Johnson Method for Kirchhoff plates": [[65, "the-hellan-herrmann-johnson-method-for-kirchhoff-plates"]], "Kirchhoff Plate equation": [[65, "kirchhoff-plate-equation"]], "Hellan-Herrmann-Johnson method": [[65, "hellan-herrmann-johnson-method"]], "Relationship between HHJ and TDNNS": [[66, "relationship-between-hhj-and-tdnns"]], "Commuting diagram for H^1 - H(\\opcurl)": [[66, "commuting-diagram-for-h-1-h-opcurl"]], "Reissner Mindlin Plates": [[67, "reissner-mindlin-plates"]], "Nonlinear Shells": [[68, "nonlinear-shells"]], "Preamble": [[70, "preamble"]], "Natural boundary conditions": [[71, "natural-boundary-conditions"]], "Essential boundary conditions": [[71, "essential-boundary-conditions"]], "Exercises": [[73, "exercises"], [28, "exercises"]], "Get started with Netgen-Opencascade geometric modeling": [[73, "get-started-with-netgen-opencascade-geometric-modeling"]], "Experiment with CoefficientFunctions": [[73, "experiment-with-coefficientfunctions"]], "Work with GridFunctions": [[73, "work-with-gridfunctions"]], "Linear Algebra": [[73, "linear-algebra"]], "Experiments with BilienarForms and LinearForms": [[73, "experiments-with-bilienarforms-and-linearforms"]], "Computing dual norms": [[73, "computing-dual-norms"]], "Solving the Poisson Equation": [[74, "solving-the-poisson-equation"]], "Weak formulation": [[74, "weak-formulation"]], "The Sobolev space H^1, linear and bilinear forms": [[74, "the-sobolev-space-h-1-linear-and-bilinear-forms"]], "The Finite Element Method": [[74, "the-finite-element-method"]], "Poisson equation in NGSolve:": [[74, "poisson-equation-in-ngsolve"]], "Visualizing the basis functions": [[74, "visualizing-the-basis-functions"]], "Variable Coefficients": [[76, "variable-coefficients"]], "hp - Finite Elements": [[77, "hp-finite-elements"]], "Legendre Polynomials": [[77, "legendre-polynomials"]], "Error estimate of the L_2 projection": [[77, "error-estimate-of-the-l-2-projection"]], "Orthogonal polynomials on triangles": [[77, "orthogonal-polynomials-on-triangles"]], "Projection based interpolation": [[77, "projection-based-interpolation"]], "The 1D case": [[77, "the-1d-case"]], "Projection based interpolation on triangles": [[77, "projection-based-interpolation-on-triangles"]], "The Bramble-Pasciak Transformation": [[78, "the-bramble-pasciak-transformation"]], "A Small Number of Constraints": [[79, "a-small-number-of-constraints"]], "Projected preconditioner": [[79, "projected-preconditioner"]], "Dirichlet boundary conditions by penalty:": [[80, "dirichlet-boundary-conditions-by-penalty"]], "Penalty formulation for the flux:": [[80, "penalty-formulation-for-the-flux"]], "Maxwell equations:": [[80, "maxwell-equations"]], "Penalty formulation for the Stokes equation:": [[80, "penalty-formulation-for-the-stokes-equation"]], "Robust two-level methods for parameter dependent problems": [[80, "robust-two-level-methods-for-parameter-dependent-problems"]], "Robust smoothers": [[80, "robust-smoothers"]], "Robust coarse-grid correction": [[80, "robust-coarse-grid-correction"]], "Two-level analysis for Maxwell equations": [[80, "two-level-analysis-for-maxwell-equations"]], "Structure of Saddle-point Problems": [[81, "structure-of-saddle-point-problems"]], "Constrained minimization problems": [[81, "constrained-minimization-problems"]], "Examples": [[81, "examples"], [104, "examples"]], "Schur complement iteration": [[81, "schur-complement-iteration"]], "Block-preconditioning": [[81, "block-preconditioning"]], "Finite Element Error Analysis": [[83, "finite-element-error-analysis"]], "Local post-processing": [[83, "local-post-processing"]], "Error Analysis in L_2 \\times H^1": [[84, "error-analysis-in-l-2-times-h-1"]], "Finite Elements in H(\\operatorname{div})": [[85, "finite-elements-in-h-operatorname-div"]], "Piola Transformation": [[85, "piola-transformation"]], "Application of the abstract theory": [[86, "application-of-the-abstract-theory"]], "Dual mixed formulation": [[86, "dual-mixed-formulation"]], "Primal mixed formulation": [[86, "primal-mixed-formulation"]], "The function space H(\\operatorname{div})": [[87, "the-function-space-h-operatorname-div"]], "Normal-trace of functions in H(\\opdiv)": [[87, "normal-trace-of-functions-in-h-opdiv"]], "H(\\opdiv) on sub-domains": [[87, "h-opdiv-on-sub-domains"]], "Hybridization Techniques": [[88, "hybridization-techniques"]], "Friedrichs\u2019 Inequality": [[89, "friedrichs-inequality"], [92, "friedrichs-inequality"]], "The Poincar\u00e9 inequality": [[93, "the-poincare-inequality"]], "The Trace Inequality": [[94, "the-trace-inequality"]], "Numerical verification": [[94, "numerical-verification"]], "Proof of the trace inequality:": [[94, "proof-of-the-trace-inequality"]], "Generalized derivatives": [[95, "generalized-derivatives"]], "Sobolev spaces": [[96, "sobolev-spaces"]], "Trace theorems and their applications": [[97, "trace-theorems-and-their-applications"]], "Integration by parts": [[97, "integration-by-parts"]], "Sobolev spaces over sub-domains": [[97, "sobolev-spaces-over-sub-domains"]], "Extension operators": [[97, "extension-operators"]], "The trace space H^{1/2}": [[97, "the-trace-space-h-1-2"]], "Equivalent norms on H^1 and on sub-spaces": [[98, "equivalent-norms-on-h-1-and-on-sub-spaces"]], "Additive Schwarz Methods": [[99, "additive-schwarz-methods"]], "Block-Jacobi and general additive Schwarz preconditioners": [[99, "block-jacobi-and-general-additive-schwarz-preconditioners"]], "Sub-spaces of {\\mathbb R}^N and of Hilbert-spaces": [[99, "sub-spaces-of-mathbb-r-n-and-of-hilbert-spaces"]], "The Additive Schwarz Lemma": [[99, "the-additive-schwarz-lemma"]], "The upper-bound by the overlap": [[99, "the-upper-bound-by-the-overlap"]], "Some Examples of ASM preconditioners": [[100, "some-examples-of-asm-preconditioners"]], "Diagonal preconditioner for L_2-norm": [[100, "diagonal-preconditioner-for-l-2-norm"]], "Diagonal preconditioner for the H^1 norm": [[100, "diagonal-preconditioner-for-the-h-1-norm"]], "H^1-norm with small L_2-term": [[100, "h-1-norm-with-small-l-2-term"]], "Exercise: Robust preconditioners": [[101, "exercise-robust-preconditioners"]], "Exercise A:": [[101, "exercise-a"]], "Exercise B:": [[101, "exercise-b"]], "Domain Decomposition with minimal overlap": [[102, "domain-decomposition-with-minimal-overlap"]], "Analysis of the method": [[102, "analysis-of-the-method"]], "Adding a coarse grid space": [[102, "adding-a-coarse-grid-space"]], "Graph-based mesh partitioning": [[102, "graph-based-mesh-partitioning"]], "Overlapping Domain Decomposition Methods": [[103, "overlapping-domain-decomposition-methods"]], "Experiments with overlapping DD": [[103, "experiments-with-overlapping-dd"]], "Analysis of the DD preconditioner": [[103, "analysis-of-the-dd-preconditioner"]], "Overlapping DD Methods with coarse grid": [[103, "overlapping-dd-methods-with-coarse-grid"]], "Analysis of the 2-level method:": [[103, "analysis-of-the-2-level-method"]], "Comparison to DD with minimal overlap": [[103, "comparison-to-dd-with-minimal-overlap"]], "Runge Kutta Methods": [[104, "runge-kutta-methods"]], "Butcher tableaus of simple methods": [[104, "butcher-tableaus-of-simple-methods"]], "Convergence of Runge Kutta methods": [[104, "convergence-of-runge-kutta-methods"]], "Explicit methods:": [[104, "explicit-methods"]], "Diagonally implicit Runge-Kutta methods:": [[104, "diagonally-implicit-runge-kutta-methods"]], "Exercise:": [[104, "exercise"], [114, "exercise"]], "Single-step methods": [[105, "single-step-methods"]], "Simple methods based on the integral equation": [[105, "simple-methods-based-on-the-integral-equation"]], "Explicit Euler method (EE)": [[105, "explicit-euler-method-ee"]], "Implicit Euler method (IE)": [[105, "implicit-euler-method-ie"]], "Trapezoidal method": [[105, "trapezoidal-method"]], "Explicit mid-point rule (=improved Euler method = RK2 method)": [[105, "explicit-mid-point-rule-improved-euler-method-rk2-method"]], "Linear stability classification": [[105, "linear-stability-classification"]], "Stability function": [[105, "stability-function"]], "Stability conditions": [[105, "stability-conditions"]], "Single step methods and parabolic equations": [[105, "single-step-methods-and-parabolic-equations"]], "Single step methods and hyperbolic equations": [[105, "single-step-methods-and-hyperbolic-equations"]], "Exercises:": [[105, "exercises"]], "Heat Equation": [[106, "heat-equation"]], "Variational formulation in space": [[106, "variational-formulation-in-space"]], "Galerkin method in space": [[106, "galerkin-method-in-space"]], "Implicit Euler time-stepping": [[106, "implicit-euler-time-stepping"]], "Wave Equation": [[107, "wave-equation"]], "Newmark time-stepping method": [[107, "newmark-time-stepping-method"]], "Exponential Integrators for Parabolic Equations": [[108, "exponential-integrators-for-parabolic-equations"]], "Mass-lumping and Local time-stepping": [[111, "mass-lumping-and-local-time-stepping"]], "The Verlet method:": [[111, "the-verlet-method"]], "Geometry with local details:": [[111, "geometry-with-local-details"]], "Nano-optics: A ring-resonator": [[112, "nano-optics-a-ring-resonator"], [113, "nano-optics-a-ring-resonator"]], "Discontinuous Galerkin for the Wave Equation": [[114, "discontinuous-galerkin-for-the-wave-equation"]], "Testing the differential operators": [[114, "testing-the-differential-operators"]], "Efficient implementation:": [[114, "efficient-implementation"]], "Solving the wave equation:": [[114, "solving-the-wave-equation"]], "Eigenvalues of the discretized Laplace-operator": [[114, "eigenvalues-of-the-discretized-laplace-operator"]], "Coercive variational problems and their approximation": [[26, "coercive-variational-problems-and-their-approximation"]], "Approximation of coercive variational problems": [[26, "approximation-of-coercive-variational-problems"]], "Inf-sup stable variational problems": [[29, "inf-sup-stable-variational-problems"]], "Approximation of inf-sup stable variational problems": [[29, "approximation-of-inf-sup-stable-variational-problems"]], "3D Solid Mechanics": [[72, "d-solid-mechanics"]], "Linear elasticity": [[72, "linear-elasticity"]], "Variational formulation:": [[72, "variational-formulation"]], "Minimization problem": [[28, "minimization-problem"]], "inf-sup condition of the first-order derivative bilinear-form": [[28, "inf-sup-condition-of-the-first-order-derivative-bilinear-form"]], "Repeat the exercise in 2D.": [[28, "repeat-the-exercise-in-2d"]], "Building systems from building-blocks": [[28, "building-systems-from-building-blocks"]], "Coercive examples": [[28, "coercive-examples"]], "inf-sup condition": [[28, "inf-sup-condition"]], "complex-valued problem as real system": [[28, "complex-valued-problem-as-real-system"]], "One sup is enough": [[28, "one-sup-is-enough"]], "Second inf-sup condition means onto": [[28, "second-inf-sup-condition-means-onto"]], "Basic properties": [[25, "basic-properties"]], "Projection onto subspaces": [[30, "projection-onto-subspaces"]], "Non-linear dynamics": [[57, "non-linear-dynamics"]], "3D-TDNNS Elasticity and Reissner Mindlin Plate elements": [[69, "d-tdnns-elasticity-and-reissner-mindlin-plate-elements"]]}, "indexentries": {}}) \ No newline at end of file +Search.setIndex({"docnames": ["DG/Nitsche", "DG/elliptic", "DG/elliptic_stdDG", "DG/fourthorder", "DG/instationary", "DG/splitting", "DG/stationary", "DG/stokes", "FEM/erroranalysis", "FEM/finiteelements", "FEM/myFiniteElement", "FEM/nonconforming", "FEM/sytemassembling", 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"FEM/finiteelements.ipynb", "FEM/myFiniteElement.ipynb", "FEM/nonconforming.ipynb", "FEM/sytemassembling.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", "abstracttheory/BasicProperties.ipynb", "abstracttheory/Coercive.ipynb", "abstracttheory/RieszRepresentation.ipynb", "abstracttheory/exercises.ipynb", "abstracttheory/infsup.ipynb", "abstracttheory/subspaceprojection.ipynb", "aposteriori/aposteriori.ipynb", "aposteriori/equilibrated.ipynb", "aposteriori/goaldriven.ipynb", "aposteriori/residualEE.ipynb", "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/algorithms.ipynb", "multigrid/analysisMG.ipynb", "multigrid/analysisML.ipynb", "multigrid/multilevel-extension.ipynb", "plates/hhj.ipynb", "plates/hhj-tdnns.ipynb", "plates/reissnermindlin.ipynb", "plates/shells.ipynb", "plates/tdnnsplate.ipynb", "preamble.ipynb", 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"subspacecorrection/minimaldd.ipynb", "subspacecorrection/overlapping.ipynb", "timedependent/ODEs/runge_kutta.ipynb", "timedependent/ODEs/singlestep.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": ["35. Nitsche\u2019s Method for boundary and interface conditions", "37. Hybrid DG for elliptic equations", "DG - Methods for elliptic problems", "39. Fourth Order Equation", "34. Instationary Transport Equation", "38. Splitting Methods for the time-dependent convection diffusion equation", "33. Stationary Transport Equation", "40. H(div)-conforming Stokes", "19. Finite element error analysis", "17. Finite Element Method", "21. Implementation of Finite Elements", "20. Non-conforming Finite Element Methods", "18. Finite element system assembling", "80. NGSolve - PETSc interface", "75. Distributed Meshes and Spaces", "<no title>", "<no title>", "<no title>", "<no title>", "74. Introduction to MPI with mpi4py", "78. Iteration methods in parallel", "81. Solving Stokes in parallel", "79. Using PETSc", "77. Consistent and Distributed Vectors", "This repository contains an interactive introduction to the Finite Element Method", "7. Basic properties", "10. Coercive variational problems and their approximation", "9. Riesz representation theorem and symmetric variational problems", "12. Exercises", "11. Inf-sup stable variational problems", "8. Projection onto subspaces", "22. A posteriori error estimates", "25. Equilibrated Residual Error Estimates", "24. Goal driven error estimates", "23. The residual error estimator", "73. BDDC - Preconditioner", "71. FETI methods", "72. FETI-DP", "68. Introduction to Non-overlapping Domain Decomposition", "69. 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", "51. The Chebyshev Method", "48. The Richardson Iteration", "52. Conjugate Gradients", "49. The Gradient Method", "50. Preconditioning", "47. Basic Iterative Methods", "30. Abstract Theory", "31. Abstract theory for mixed finite element methods", "28. Boundary Conditions", "32. Parameter Dependent Problems", "29. Mixed Methods for second order equations", "27. 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", "58. Multigrid and Multilevel Methods", "60. Analysis of the Multigrid Iteration", "59. Analysis of the multi-level preconditioner", "61. Multi-level Extension", "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", "2. Boundary Conditions", "5. 3D Solid Mechanics", "6. Exercises", "1. Solving the Poisson Equation", "4. Iterative Solvers", "3. Variable Coefficients", "26. hp - Finite Elements", "63. The Bramble-Pasciak Transformation", "64. A Small Number of Constraints", "66. Parameter Dependent Problems", "62. Structure of Saddle-point Problems", "<no title>", "44. Finite Element Error Analysis", "45. Error Analysis in \\(L_2 \\times H^1\\)", "43. Finite Elements in \\(H(\\operatorname{div})\\)", "41. Application of the abstract theory", "42. The function space \\(H(\\operatorname{div})\\)", "46. Hybridization Techniques", "Friedrichs\u2019 Inequality", "<no title>", "<no title>", "Friedrichs\u2019 Inequality", "The Poincar\u00e9 inequality", "The Trace Inequality", "13. Generalized derivatives", "14. Sobolev spaces", "15. Trace theorems and their applications", "16. Equivalent norms on \\(H^1\\) and on sub-spaces", "53. Additive Schwarz Methods", "54. Some Examples of ASM preconditioners", "57. Exercise: Robust preconditioners", "55. Domain Decomposition with minimal overlap", "56. 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51, 58, 65, 71, 73, 74, 76, 79, 80, 81, 83, 84, 88, 98, 99, 107], "problem": [8, 10, 11, 12, 20, 32, 33, 35, 36, 37, 38, 39, 45, 48, 49, 58, 61, 62, 63, 65, 67, 68, 73, 74, 75, 77, 79, 83, 86, 88, 94, 97, 98, 99, 100, 101, 102, 103, 105], "sub": [8, 11, 22, 23, 25, 28, 30, 32, 35, 36, 37, 38, 40, 47, 51, 52, 61, 63, 71, 74, 75, 77, 79, 80, 81, 85, 95, 100, 102, 103, 104, 105, 106], "cea": [8, 11, 62], "": [8, 9, 11, 17, 18, 24, 25, 26, 27, 28, 30, 32, 37, 43, 44, 51, 52, 54, 62, 68, 69, 70, 72, 73, 74, 75, 79, 81, 89, 97, 98, 99, 104, 105, 108, 111, 114], "lemma": [8, 9, 25, 26, 29, 30, 34, 35, 43, 61, 62, 63, 77, 80, 97, 98, 102, 103, 105], "bound": [8, 9, 11, 25, 26, 28, 29, 31, 32, 33, 34, 45, 46, 47, 49, 63, 80, 81, 94, 95, 96, 97, 98, 100, 102, 103], "best": [8, 26, 52, 65, 77], "_v": [8, 11, 25, 26, 27, 28, 29, 31, 33, 34, 51, 52, 73, 97, 98], "leq": [8, 9, 11, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 45, 46, 47, 48, 49, 51, 52, 59, 62, 63, 64, 65, 69, 77, 80, 81, 83, 87, 89, 92, 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46, 47, 49, 51, 52, 61, 62, 64, 67, 74, 77, 80, 81, 88, 94, 95, 97, 98, 99, 103, 104, 105, 106, 114], "studi": [8, 38, 56, 73, 97], "converg": [8, 25, 26, 29, 30, 31, 36, 37, 39, 46, 48, 49, 54, 63, 73, 75, 77, 80, 96, 97, 98, 108], "famili": [8, 85], "decreas": [8, 38, 46, 49], "case": [8, 9, 12, 22, 26, 27, 28, 29, 33, 34, 49, 52, 56, 58, 63, 65, 67, 71, 74, 76, 80, 81, 96, 97, 98, 99, 100, 108], "geq": [8, 11, 25, 26, 28, 29, 30, 31, 32, 45, 49, 51, 52, 62, 64, 65, 77, 81, 84, 97], "refer": [8, 9, 10, 11, 34, 50, 60, 62, 63, 85, 100, 105, 114], "triangl": [8, 9, 31, 32, 34, 52, 64, 80, 83, 100, 111], "b_t": 8, "simeq": [8, 11, 34, 97, 98], "main": [8, 32], "applic": [8, 11, 31, 33, 43, 44, 47, 61, 70, 78, 79, 114], "brambl": [8, 11, 34, 43, 44, 62, 77, 82, 98], "hilbert": [8, 11, 25, 26, 27, 28, 29, 30, 34, 51, 52, 54, 74, 77, 80, 96, 97, 98, 106], "sometim": 8, "itself": [8, 14, 19, 99, 107], "v_t": [8, 9, 23, 32, 60, 85], "psi_t": [8, 9], "contain": [8, 9, 22, 25, 31, 32, 34, 39, 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Equations": [[43, "mixed-methods-for-second-order-equations"], [44, null]], "Mixed Methods for Elasticity": [[43, "mixed-methods-for-elasticity"]], "Mixed Methods for Plates and Shells": [[43, "mixed-methods-for-plates-and-shells"]], "The Helmholtz equation": [[43, "the-helmholtz-equation"]], "Iterative Solvers": [[43, "iterative-solvers"], [75, "iterative-solvers"]], "Iteration Methods": [[43, "iteration-methods"], [44, null]], "Sub-space Correction Methods": [[43, "sub-space-correction-methods"], [44, null]], "Multigrid Methods": [[43, "multigrid-methods"], [44, null]], "Saddle-point Problems": [[43, "saddle-point-problems"], [44, null]], "Non-overlapping Domain Decomposition Methods": [[43, "non-overlapping-domain-decomposition-methods"], [44, null]], "Parallel Solvers": [[43, "parallel-solvers"], [44, null]], "Time-dependent Problems": [[43, "time-dependent-problems"]], "A practical introduction": [[43, "a-practical-introduction"]], "Ordinary differential equations": [[43, "ordinary-differential-equations"]], "Numerical analysis of Parabolic Equations": [[43, "numerical-analysis-of-parabolic-equations"]], "Numerical analysis of Wave Equations": [[43, "numerical-analysis-of-wave-equations"]], "Additional methods": [[43, "additional-methods"]], "Appendix": [[43, "appendix"]], "Some inequalities in Sobolev spaces": [[43, "some-inequalities-in-sobolev-spaces"]], "Literature": [[44, "literature"]], "Installing NGSolve": [[44, "installing-ngsolve"]], "Sobolev Spaces": [[44, null]], "High Order Finite Elements": [[44, null]], "The Chebyshev Method": [[45, "the-chebyshev-method"]], "Chebyshev polynomials": [[45, "chebyshev-polynomials"]], "The Chebyshev iteration": [[45, "the-chebyshev-iteration"]], "The Richardson Iteration": [[46, "the-richardson-iteration"]], "Optimizing the relaxation parameter \\alpha": [[46, "optimizing-the-relaxation-parameter-alpha"]], "Experiments with the Richardson iteration": [[46, "experiments-with-the-richardson-iteration"]], "Conjugate Gradients": [[47, "conjugate-gradients"]], "Solving the minimization problem": [[47, "solving-the-minimization-problem"]], "Expanding the Krylov-space": [[47, "expanding-the-krylov-space"]], "The Gradient Method": [[48, "the-gradient-method"]], "Preconditioning": [[49, "preconditioning"]], "The preconditioned Richardson iteration": [[49, "the-preconditioned-richardson-iteration"]], "The preconditioned gradient method": [[49, "the-preconditioned-gradient-method"]], "Jacobi and Gauss Seidel Preconditioners": [[49, "jacobi-and-gauss-seidel-preconditioners"]], "Basic Iterative Methods": [[50, "basic-iterative-methods"]], "Constrained minimization problem": [[51, "constrained-minimization-problem"]], "Stokes equation within the abstract theory": [[51, "stokes-equation-within-the-abstract-theory"]], "Dirichlet boundary conditions as mixed system": [[51, "dirichlet-boundary-conditions-as-mixed-system"]], "Mixed method for second order equation": [[51, "mixed-method-for-second-order-equation"]], "Abstract theory for mixed finite element methods": [[52, "abstract-theory-for-mixed-finite-element-methods"]], "Finite element error estimates": [[52, "finite-element-error-estimates"]], "Proving discrete LBB by the Fortin operator": [[52, "proving-discrete-lbb-by-the-fortin-operator"]], "Example: Finite elements for Stokes": [[52, "example-finite-elements-for-stokes"]], "Boundary Conditions": [[53, "boundary-conditions"], [71, "boundary-conditions"]], "Parameter Dependent Problems": [[54, "parameter-dependent-problems"], [80, "parameter-dependent-problems"]], "Example: Dirichlet boundary condition by penalty": [[54, "example-dirichlet-boundary-condition-by-penalty"]], "Example: Nearly incompressible materials": [[54, "example-nearly-incompressible-materials"]], "Mixed Methods for second order equations": [[55, "mixed-methods-for-second-order-equations"]], "Stokes Equation": [[56, "stokes-equation"]], "Variational Formulation": [[56, "variational-formulation"]], "Finite Element Spaces": [[56, "finite-element-spaces"]], "\\DeclareMathOperator{\\opdiv}{div}\nHellinger Reissner mixed formulation": [[58, "declaremathoperator-opdiv-div-hellinger-reissner-mixed-formulation"]], "Primal mixed method:": [[58, "primal-mixed-method"]], "Dual mixed method": [[58, "dual-mixed-method"]], "\\DeclareMathOperator{\\opdiv}{div}\n\\DeclareMathOperator{\\opcurl}{curl}\n\\DeclareMathOperator{\\eps}{\\varepsilon}\nReduced symmetry methods": [[59, "declaremathoperator-opdiv-div-declaremathoperator-opcurl-curl-declaremathoperator-eps-varepsilon-reduced-symmetry-methods"]], "Tangential displacement normal normal stress continuous finite elements": [[60, "tangential-displacement-normal-normal-stress-continuous-finite-elements"]], "The divergence of nn-continuous piece-wise smooth functions:": [[60, "the-divergence-of-nn-continuous-piece-wise-smooth-functions"]], "TDNNS Variational formulation:": [[60, "tdnns-variational-formulation"]], "Error estimates:": [[60, "error-estimates"]], "Multigrid and Multilevel Methods": [[61, "multigrid-and-multilevel-methods"]], "Multilevel preconditioner": [[61, "multilevel-preconditioner"]], "Multigrid Preconditioning": [[61, "multigrid-preconditioning"]], "Projection matrices from the finest level": [[61, "projection-matrices-from-the-finest-level"]], "Analysis of the Multigrid Iteration": [[62, "analysis-of-the-multigrid-iteration"]], "The Algorithm": [[62, "the-algorithm"]], "The Approximation Property": [[62, "the-approximation-property"]], "The Smothing Property": [[62, "the-smothing-property"]], "Optimal convergence of the V-cycle": [[62, "optimal-convergence-of-the-v-cycle"]], "Analysis of the multi-level preconditioner": [[63, "analysis-of-the-multi-level-preconditioner"]], "Nearly optimal analysis of the ML - preconditioner": [[63, "nearly-optimal-analysis-of-the-ml-preconditioner"]], "Optimal analysis of the multi-level preconditioner": [[63, "optimal-analysis-of-the-multi-level-preconditioner"]], "Multi-level Extension": [[64, "multi-level-extension"]], "Efficiently computable multi-level decomposition": [[64, "efficiently-computable-multi-level-decomposition"]], "Algorithm": [[64, "algorithm"]], "Extending boundary data": [[64, "extending-boundary-data"]], "The Hellan Herrmann Johnson Method for Kirchhoff plates": [[65, "the-hellan-herrmann-johnson-method-for-kirchhoff-plates"]], "Kirchhoff Plate equation": [[65, "kirchhoff-plate-equation"]], "Hellan-Herrmann-Johnson method": [[65, "hellan-herrmann-johnson-method"]], "Relationship between HHJ and TDNNS": [[66, "relationship-between-hhj-and-tdnns"]], "Commuting diagram for H^1 - H(\\opcurl)": [[66, "commuting-diagram-for-h-1-h-opcurl"]], "Reissner Mindlin Plates": [[67, "reissner-mindlin-plates"]], "Nonlinear Shells": [[68, "nonlinear-shells"]], "Preamble": [[70, "preamble"]], "Natural boundary conditions": [[71, "natural-boundary-conditions"]], "Essential boundary conditions": [[71, "essential-boundary-conditions"]], "Exercises": [[73, "exercises"], [28, "exercises"]], "Get started with Netgen-Opencascade geometric modeling": [[73, "get-started-with-netgen-opencascade-geometric-modeling"]], "Experiment with CoefficientFunctions": [[73, "experiment-with-coefficientfunctions"]], "Work with GridFunctions": [[73, "work-with-gridfunctions"]], "Linear Algebra": [[73, "linear-algebra"]], "Experiments with BilienarForms and LinearForms": [[73, "experiments-with-bilienarforms-and-linearforms"]], "Computing dual norms": [[73, "computing-dual-norms"]], "Solving the Poisson Equation": [[74, "solving-the-poisson-equation"]], "Weak formulation": [[74, "weak-formulation"]], "The Sobolev space H^1, linear and bilinear forms": [[74, "the-sobolev-space-h-1-linear-and-bilinear-forms"]], "The Finite Element Method": [[74, "the-finite-element-method"]], "Poisson equation in NGSolve:": [[74, "poisson-equation-in-ngsolve"]], "Visualizing the basis functions": [[74, "visualizing-the-basis-functions"]], "Variable Coefficients": [[76, "variable-coefficients"]], "hp - Finite Elements": [[77, "hp-finite-elements"]], "Legendre Polynomials": [[77, "legendre-polynomials"]], "Error estimate of the L_2 projection": [[77, "error-estimate-of-the-l-2-projection"]], "Orthogonal polynomials on triangles": [[77, "orthogonal-polynomials-on-triangles"]], "Projection based interpolation": [[77, "projection-based-interpolation"]], "The 1D case": [[77, "the-1d-case"]], "Projection based interpolation on triangles": [[77, "projection-based-interpolation-on-triangles"]], "The Bramble-Pasciak Transformation": [[78, "the-bramble-pasciak-transformation"]], "A Small Number of Constraints": [[79, "a-small-number-of-constraints"]], "Projected preconditioner": [[79, "projected-preconditioner"]], "Dirichlet boundary conditions by penalty:": [[80, "dirichlet-boundary-conditions-by-penalty"]], "Penalty formulation for the flux:": [[80, "penalty-formulation-for-the-flux"]], "Maxwell equations:": [[80, "maxwell-equations"]], "Penalty formulation for the Stokes equation:": [[80, "penalty-formulation-for-the-stokes-equation"]], "Robust two-level methods for parameter dependent problems": [[80, "robust-two-level-methods-for-parameter-dependent-problems"]], "Robust smoothers": [[80, "robust-smoothers"]], "Robust coarse-grid correction": [[80, "robust-coarse-grid-correction"]], "Two-level analysis for Maxwell equations": [[80, "two-level-analysis-for-maxwell-equations"]], "Structure of Saddle-point Problems": [[81, "structure-of-saddle-point-problems"]], "Constrained minimization problems": [[81, "constrained-minimization-problems"]], "Examples": [[81, "examples"], [104, "examples"]], "Schur complement iteration": [[81, "schur-complement-iteration"]], "Block-preconditioning": [[81, "block-preconditioning"]], "Finite Element Error Analysis": [[83, "finite-element-error-analysis"]], "Local post-processing": [[83, "local-post-processing"]], "Error Analysis in L_2 \\times H^1": [[84, "error-analysis-in-l-2-times-h-1"]], "Finite Elements in H(\\operatorname{div})": [[85, "finite-elements-in-h-operatorname-div"]], "Piola Transformation": [[85, "piola-transformation"]], "Application of the abstract theory": [[86, "application-of-the-abstract-theory"]], "Dual mixed formulation": [[86, "dual-mixed-formulation"]], "Primal mixed formulation": [[86, "primal-mixed-formulation"]], "The function space H(\\operatorname{div})": [[87, "the-function-space-h-operatorname-div"]], "Normal-trace of functions in H(\\opdiv)": [[87, "normal-trace-of-functions-in-h-opdiv"]], "H(\\opdiv) on sub-domains": [[87, "h-opdiv-on-sub-domains"]], "Hybridization Techniques": [[88, "hybridization-techniques"]], "Friedrichs\u2019 Inequality": [[89, "friedrichs-inequality"], [92, "friedrichs-inequality"]], "The Poincar\u00e9 inequality": [[93, "the-poincare-inequality"]], "The Trace Inequality": [[94, "the-trace-inequality"]], "Numerical verification": [[94, "numerical-verification"]], "Proof of the trace inequality:": [[94, "proof-of-the-trace-inequality"]], "Generalized derivatives": [[95, "generalized-derivatives"]], "Sobolev spaces": [[96, "sobolev-spaces"]], "Trace theorems and their applications": [[97, "trace-theorems-and-their-applications"]], "Integration by parts": [[97, "integration-by-parts"]], "Sobolev spaces over sub-domains": [[97, "sobolev-spaces-over-sub-domains"]], "Extension operators": [[97, "extension-operators"]], "The trace space H^{1/2}": [[97, "the-trace-space-h-1-2"]], "Equivalent norms on H^1 and on sub-spaces": [[98, "equivalent-norms-on-h-1-and-on-sub-spaces"]], "Additive Schwarz Methods": [[99, "additive-schwarz-methods"]], "Block-Jacobi and general additive Schwarz preconditioners": [[99, "block-jacobi-and-general-additive-schwarz-preconditioners"]], "Sub-spaces of {\\mathbb R}^N and of Hilbert-spaces": [[99, "sub-spaces-of-mathbb-r-n-and-of-hilbert-spaces"]], "The Additive Schwarz Lemma": [[99, "the-additive-schwarz-lemma"]], "The upper-bound by the overlap": [[99, "the-upper-bound-by-the-overlap"]], "Some Examples of ASM preconditioners": [[100, "some-examples-of-asm-preconditioners"]], "Diagonal preconditioner for L_2-norm": [[100, "diagonal-preconditioner-for-l-2-norm"]], "Diagonal preconditioner for the H^1 norm": [[100, "diagonal-preconditioner-for-the-h-1-norm"]], "H^1-norm with small L_2-term": [[100, "h-1-norm-with-small-l-2-term"]], "Exercise: Robust preconditioners": [[101, "exercise-robust-preconditioners"]], "Exercise A:": [[101, "exercise-a"]], "Exercise B:": [[101, "exercise-b"]], "Domain Decomposition with minimal overlap": [[102, "domain-decomposition-with-minimal-overlap"]], "Analysis of the method": [[102, "analysis-of-the-method"]], "Adding a coarse grid space": [[102, "adding-a-coarse-grid-space"]], "Graph-based mesh partitioning": [[102, "graph-based-mesh-partitioning"]], "Overlapping Domain Decomposition Methods": [[103, "overlapping-domain-decomposition-methods"]], "Experiments with overlapping DD": [[103, "experiments-with-overlapping-dd"]], "Analysis of the DD preconditioner": [[103, "analysis-of-the-dd-preconditioner"]], "Overlapping DD Methods with coarse grid": [[103, "overlapping-dd-methods-with-coarse-grid"]], "Analysis of the 2-level method:": [[103, "analysis-of-the-2-level-method"]], "Comparison to DD with minimal overlap": [[103, "comparison-to-dd-with-minimal-overlap"]], "Runge Kutta Methods": [[104, "runge-kutta-methods"]], "Butcher tableaus of simple methods": [[104, "butcher-tableaus-of-simple-methods"]], "Convergence of Runge Kutta methods": [[104, "convergence-of-runge-kutta-methods"]], "Explicit methods:": [[104, "explicit-methods"]], "Diagonally implicit Runge-Kutta methods:": [[104, "diagonally-implicit-runge-kutta-methods"]], "Exercise:": [[104, "exercise"], [114, "exercise"]], "Single-step methods": [[105, "single-step-methods"]], "Simple methods based on the integral equation": [[105, "simple-methods-based-on-the-integral-equation"]], "Explicit Euler method (EE)": [[105, "explicit-euler-method-ee"]], "Implicit Euler method (IE)": [[105, "implicit-euler-method-ie"]], "Trapezoidal method": [[105, "trapezoidal-method"]], "Explicit mid-point rule (=improved Euler method = RK2 method)": [[105, "explicit-mid-point-rule-improved-euler-method-rk2-method"]], "Linear stability classification": [[105, "linear-stability-classification"]], "Stability function": [[105, "stability-function"]], "Stability conditions": [[105, "stability-conditions"]], "Single step methods and parabolic equations": [[105, "single-step-methods-and-parabolic-equations"]], "Single step methods and hyperbolic equations": [[105, "single-step-methods-and-hyperbolic-equations"]], "Exercises:": [[105, "exercises"]], "Heat Equation": [[106, "heat-equation"]], "Variational formulation in space": [[106, "variational-formulation-in-space"]], "Galerkin method in space": [[106, "galerkin-method-in-space"]], "Implicit Euler time-stepping": [[106, "implicit-euler-time-stepping"]], "Wave Equation": [[107, "wave-equation"]], "Newmark time-stepping method": [[107, "newmark-time-stepping-method"]], "Exponential Integrators for Parabolic Equations": [[108, "exponential-integrators-for-parabolic-equations"]], "Mass-lumping and Local time-stepping": [[111, "mass-lumping-and-local-time-stepping"]], "The Verlet method:": [[111, "the-verlet-method"]], "Geometry with local details:": [[111, "geometry-with-local-details"]], "Nano-optics: A ring-resonator": [[112, "nano-optics-a-ring-resonator"], [113, "nano-optics-a-ring-resonator"]], "Discontinuous Galerkin for the Wave Equation": [[114, "discontinuous-galerkin-for-the-wave-equation"]], "Testing the differential operators": [[114, "testing-the-differential-operators"]], "Efficient implementation:": [[114, "efficient-implementation"]], "Solving the wave equation:": [[114, "solving-the-wave-equation"]], "Eigenvalues of the discretized Laplace-operator": [[114, "eigenvalues-of-the-discretized-laplace-operator"]], "Coercive variational problems and their approximation": [[26, "coercive-variational-problems-and-their-approximation"]], "Approximation of coercive variational problems": [[26, "approximation-of-coercive-variational-problems"]], "3D Solid Mechanics": [[72, "d-solid-mechanics"]], "Linear elasticity": [[72, "linear-elasticity"]], "Variational formulation:": [[72, "variational-formulation"]], "Minimization problem": [[28, "minimization-problem"]], "inf-sup condition of the first-order derivative bilinear-form": [[28, "inf-sup-condition-of-the-first-order-derivative-bilinear-form"]], "Repeat the exercise in 2D.": [[28, "repeat-the-exercise-in-2d"]], "Building systems from building-blocks": [[28, "building-systems-from-building-blocks"]], "Coercive examples": [[28, "coercive-examples"]], "inf-sup condition": [[28, "inf-sup-condition"]], "complex-valued problem as real system": [[28, "complex-valued-problem-as-real-system"]], "One sup is enough": [[28, "one-sup-is-enough"]], "Second inf-sup condition means onto": [[28, "second-inf-sup-condition-means-onto"]], "Basic properties": [[25, "basic-properties"]], "Projection onto subspaces": [[30, "projection-onto-subspaces"]], "Inf-sup stable variational problems": [[29, "inf-sup-stable-variational-problems"]], "Approximation of inf-sup stable variational problems": [[29, "approximation-of-inf-sup-stable-variational-problems"]], "Non-linear dynamics": [[57, "non-linear-dynamics"]], "3D-TDNNS Elasticity and Reissner Mindlin Plate elements": [[69, "d-tdnns-elasticity-and-reissner-mindlin-plate-elements"]]}, "indexentries": {}}) \ No newline at end of file