examples/navier-stokes.md

@@ -5,14 +5,15 @@ We consider the incompressible Navier-Stokes equation in a rectangular domain $`
 \varrho(\partial_t\mathbf{u} + (\mathbf{u}\cdot\nabla)\mathbf{u}) - \nabla\cdot\big(2\nu\mathbf{D}(\mathbf{u})\big) - \nabla p = \mathbf{f},\qquad\text{ in }\Omega \\
 \nabla\cdot\mathbf{u} = 0
 ```
-with velocity $`\mathbf{u}`$, pressure $`p`$, density $\varrho$, dynamic viscosity $`\nu`$, 
+with velocity $`\mathbf{u}`$, pressure $`p`$, density $`\varrho`$, dynamic viscosity $`\nu`$,
 rate-of-strain tensor $`\mathbf{D}(\mathbf{u})=\frac{1}{2}(\nabla\mathbf{u}+\nabla\mathbf{u}^\top)`$,
 and external volume force $`\mathbf{f}`$, w.r.t. to boundary conditions $`\mathbf{u}=\mathbf{g}`$ on $`\partial\Omega`$.
 For a coarse overview, see also [Wikipedia](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations).
 
+Discretization
+--------------
 
-Weak formulation
-----------------
+### 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
 ```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, \\
@@ -23,8 +24,7 @@ for $`t\in[0,T]`$, given $`\mathbf{u}(0,\cdot)\equiv\mathbf{u}^0`$.
 The pair $`\big(\mathbf{u}(t,\cdot), p(t,\cdot)\big)`$ is thus in the product space $`\mathbf{V}_\mathbf{g}\times Q`$ where $`\mathbf{V}_\mathbf{g}`$ is also a product of $`H^1`$ spaces, i.e., $`\mathbf{V}_\mathbf{g}=V_{g_0}\times V_{g_1}\times V_{g_2}\simeq [V_{g_i}]^d`$. Later, this space will be approximated with a Taylor-Hood space of piecewise quadratic and linear functions.
 
 
-Time-discretization
--------------------
+### Time-discretization
 A linearized time-discretization with a semi-implicit backward Euler scheme then reads: for $`k=0,1,2,\ldots, N`$ find $`\mathbf{u}^{k+1}\in\mathbf{V}_\mathbf{g},\, p\in Q`$, s.t.
 ```math
 \int_\Omega \varrho\Big(\frac{1}{\tau}(\mathbf{u}^{k+1} - \mathbf{u}^k) + (\mathbf{u}^k\cdot\nabla)\mathbf{u}^{k+1}\Big)\cdot\mathbf{v} + \nu\nabla\mathbf{u}^{k+1}:\nabla\mathbf{v} + p\,\nabla\cdot\mathbf{v}\,\text{d}\mathbf{x} = \int_\Omega \mathbf{f}(t^{k+1})\cdot\mathbf{v}\,\text{d}\mathbf{x},\qquad\forall \mathbf{v}\in \mathbf{V}_0, \\
@@ -33,6 +33,5 @@ A linearized time-discretization with a semi-implicit backward Euler scheme then
 with $`0=t^0<t^1<\ldots<t^N=T`$, timestep $`\tau=t^{k+1} - t^k`$, and given initial velocity $`\mathbf{u}^0`$.
 
 
-Finite-Element formulation
---------------------------
-Let $`\Omega`$ be partitioned into quasi-uniform elements by a conforming triangulation $`\mathcal{T}_h`$. The functions $`\mathbf{u}`$ and $`p`$ are approximated by functions from the discrete functionspace $`\mathbb{T}_h := [V_{2,g_i}]^d\times V_1`$ with $`V_{p,g} := \{v\in C^0(\Omega)\,:\, v|_T\in\mathbb{P}_p(T),\, T\in\mathcal{T}_h,\,\operatorname{tr}_{\partial\Omega}=g\}`$
+### Finite-Element formulation
+Let $`\Omega`$ be partitioned into quasi-uniform elements by a conforming triangulation $`\mathcal{T}_h`$. The functions $`\mathbf{u}`$ and $`p`$ are approximated by functions from the discrete functionspace $`\mathbb{T}_h := [V_{2,\,g_i}]^d\times V_1`$ with $`V_{p} := \{v\in C^0(\Omega)\,:\, v|_T\in\mathbb{P}_p(T),\, \forall T\in\mathcal{T}_h\}`$ and $`V_{p,\,g} := \{v\in V_p\,:\,\operatorname{tr}_{\partial\Omega}=g\}`$.