examples/navier-stokes.md

@@ -15,6 +15,13 @@ In a weak variational formulation, we try to find $`\mathbf{u}(t,\cdot)\in \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.
 ```
-for $`t\in[0,T]`$.
+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.
+
+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, \\
+\int_\Omega q\,\nabla\cdot\mathbf{u}^{k+1}\,\text{d}\mathbf{x} = 0,\qquad \forall q\in Q.
+```
+with $`0=t^0<t^1<t^2<\ldots<t^N=T`$ and given $`\mathbf{u}^0`$.