typoss

Signed-off-by: Umberto Zerbinati <[email protected]>

docs/src/PETScPC/oseen.py.rst

@@ -4,12 +4,12 @@ Vertex Patch smoothing for Augmented Lagrangian formulations of the Oseen proble
 In this tutorial, we will see how to use an augmented Lagrangian formulation to precondition the Oseen problem, i.e.
 
 .. math::
-   
-   \text{Given }\vec{\beta}\in \mathbb{R}^3 \text{ find } (\vec{u},p) \in [H^1_{0}(\Omega)]^d\times L^2(\Omega) \text{ s.t. }
-   
+
+   \text{Given } \vec{\beta} \in \mathbb{R}^3 \text{ find } (\vec{u}, p) \in [H^1_{0}(\Omega)]^d \times L^2(\Omega) \text{ s.t. }
+
    \begin{cases} 
-      \nu (\nabla \vec{u},\nabla \vec{v})_{L^2(\Omega)} + (\nabla\cdot \vec{v}, p)_{L^2(\Omega)} -(\nabla \vec{u}\vec{\beta},\vec{v})_{L^2(\Omega)} + (div(\vec{u}),div(\vec{v}))_{L^2(\Omega)}  = (\vec{f},\vec{v})_{L^2(\Omega)} \qquad \forall v\in H^1_{0}(\Omega)\\
-      (\nabla\cdot \vec{u},q)_{L^2(\Omega)} = 0 \qquad \froall q\in L^2(\Omega)
+      \nu (\nabla \vec{u}, \nabla \vec{v})_{L^2(\Omega)} + (\nabla \cdot \vec{v}, p)_{L^2(\Omega)} - (\nabla \vec{u} \vec{\beta}, \vec{v})_{L^2(\Omega)} + \gamma (\text{div}(\vec{u}), \text{div}(\vec{v}))_{L^2(\Omega)} = (\vec{f}, \vec{v})_{L^2(\Omega)} \quad \forall v \in H^1_{0}(\Omega) \\
+      (\nabla \cdot \vec{u}, q)_{L^2(\Omega)} = 0 \quad \forall q \in L^2(\Omega)
    \end{cases}
 
 Let us begin defining the parameters of the problem. ::

docs/src/PETScPC/stokes.py.rst

@@ -10,7 +10,7 @@ In particular, we will consider a Bernardi-Raugel inf-sup stable discretization
    
    \begin{cases} 
       (\nabla \vec{u},\nabla \vec{v})_{L^2(\Omega)} + (\nabla\cdot \vec{v}, p)_{L^2(\Omega)}  = (\vec{f},\vec{v})_{L^2(\Omega)} \qquad \forall v\in H^1_{0}(\Omega)\\
-      (\nabla\cdot \vec{u},q)_{L^2(\Omega)} = 0 \qquad \froall q\in L^2(\Omega)
+      (\nabla\cdot \vec{u},q)_{L^2(\Omega)} = 0 \qquad \forall q\in L^2(\Omega)
    \end{cases}
 
 Such a discretization can easily be constructed using NGSolve as follows: ::

docs/src/PETScSNES/hyperelasticity.py.rst

@@ -47,6 +47,9 @@ A discretization of this energy leads to a non-linear problem that we solve usin
     a += Variation(NeoHooke(C(u)).Compile()*dx)
     a += ((Id(3)+Grad(u.Trace()))*force)*v*ds("top")
 
+Once we have defined the energy and the weak form, we can solve the non-linear problem using `PETSc SNES`.
+In particular, we will use a Newton method with line search, and precondition the linear solves with a direct solver. ::
+
     from ngsPETSc import NonLinearSolver
     gfu_petsc = GridFunction(fes)
     gfu_ngs = GridFunction(fes)
@@ -89,4 +92,3 @@ We compare the performance of the two solvers, in the following table:
      - 10
 
 This suggests that while NGS non-linear solver when finely tuned performs as well as PETSc SNES, it is more sensitive to the choice of the damping factor. In this case, a damping factor of 0.3 was found to be the best choice.
-"""

docs/src/conf.py

@@ -26,7 +26,7 @@
 
 templates_path = ['_templates']
 exclude_patterns = ['_build', 'Thumbs.db', '.DS_Store']
-
+mathjax3_config = {'chtml': {'displayAlign': 'left'}}
 html_sourcelink_suffix = ''
 nbsphinx_prolog = r"""
 {% set docname = env.doc2path(env.docname, base='').replace('i-tutorials/', '') %}
@@ -64,4 +64,4 @@
 # https://www.sphinx-doc.org/en/master/usage/configuration.html#options-for-html-output
 
 html_theme = 'sphinx_rtd_theme'
-html_static_path = ['../_static']
+html_static_path = ['../_static']