Commit 645e6786 authored by Praetorius, Simon's avatar Praetorius, Simon
Browse files

update the links and some latex in the navier-stokes example

parent 38651101
......@@ -15,7 +15,7 @@ Discretization
--------------
### Weak formulation
In a weak variational formulation, we try to find $`\mathbf{u}(t,\cdot)\in \mathbf{V}_\mathbf{g}:=\{\mathbf{v}\in H^1(\Omega)^d\,:\, \operatorname{tr}_{\partial\Omega}\mathbf{v} = \mathbf{g}\}`$ and $`p(t,\cdot)\in Q:=L^2(\Omega)`$, such that
In a weak variational formulation, we try to find $`\mathbf{u}(t,\cdot)\in \mathbf{V}_\mathbf{g}:=\{\mathbf{v}\in H^1(\Omega)^d\,:\, \operatorname{tr}_{\partial\Omega}\mathbf{v} = \mathbf{g}\}`$ and $`p(t,\cdot)\in Q:=L_0^2(\Omega)`$, such that
```math
\int_\Omega \varrho(\partial_t\mathbf{u} + (\mathbf{u}\cdot\nabla)\mathbf{u})\cdot\mathbf{v} + \nu\nabla\mathbf{u}:\nabla\mathbf{v} + p\,\nabla\cdot\mathbf{v}\,\text{d}\mathbf{x} = \int_\Omega \mathbf{f}\cdot\mathbf{v}\,\text{d}\mathbf{x},\qquad\forall \mathbf{v}\in \mathbf{V}_0, \\
\int_\Omega q\,\nabla\cdot\mathbf{u}\,\text{d}\mathbf{x} = 0,\qquad \forall q\in Q.
......@@ -50,7 +50,7 @@ auto basis = composite(
lagrange<1>(), flatLexicographic());
```
At this point, any Dirichlet boundary conditions are ignored in the definition of the bases. We
build the Taylor-Hood basis as a composition of a product of $`d`$ $`\mathbb{V}_2`$ bases for
build the Taylor-Hood basis as a composition of a product of $`d\times\mathbb{V}_2`$ bases for
the velocity components and a $`\mathbb{V}_1`$ basis for the pressure. We have to use
`composite(...)` to combine the different types for velocity and pressure space, while we
can use `power<d>(...)` for the `d` velocity component spaces of the same type.
......@@ -76,10 +76,10 @@ with a [`ProblemStat`](../reference/Problem.md#class-problemstat) .
```c++
using namespace AMDiS;
ProblemStat prob("stokes", grid, basis);
ProblemStat prob{"stokes", grid, basis};
prob.initialize(INIT_ALL);
ProblemInstat probInstat("stokes", prob);
ProblemInstat probInstat{"stokes", prob};
probInstat.initialize(INIT_UH_OLD);
```
For convenience, we use class template argument deduction here.
......@@ -200,7 +200,7 @@ adapt.adapt();
Complete Example
----------------
The full source code of the Navier-Stokes example can be found in the repository
at [examples/navier_stokes.cc](https://gitlab.mn.tu-dresden.de/amdis/amdis-core/blob/master/examples/navier-stokes.cpp).
at [examples/navier_stokes.cc](https://gitlab.mn.tu-dresden.de/amdis/amdis-core/blob/master/examples/navier_stokes.cc).
Compile with
```bash
......@@ -214,4 +214,4 @@ and run with
The flow field with `density=1`, `viscosity=1` and `boundary velocity=10` will look like
<img src="../images/navier-stokes.png" with="50%" />
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<img src="../../images/navier-stokes.png" with="50%" />
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