examples/navier-stokes.md

@@ -10,9 +10,11 @@ rate-of-strain tensor $`\mathbf{D}(\mathbf{u})=\frac{1}{2}(\nabla\mathbf{u}+\nab
 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).
 
-In a weak variational formulation, we try to find $`\mathbf{u}(t,\cdot)\in V_\mathbf{g}:=\{\mathbf{v}\in H^1(\Omega)^d\,:\, \operatorname{tr}_{\partial\Omega}\mathbf{v} = \mathbf{g}\}`$ 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 $`p(t,\cdot)\in Q:=L^2(\Omega)`$, 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 V_0, \\
+\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 $`t\in[0,T]`$.
+
+The pair $`\big(\mathbf{u}(t,\cdot), p(t,\cdot)\big)`$ is thus in the product space $`\mathbf{V}_\mathbf{g}\times L^2(\Omega)`$ where $`\mathbf{V}_\mathbf{g}`$ is also a product space 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.