From: salgado <salgado@0785d39b-7218-0410-832d-ea1e28bc413d>
Date: Tue, 15 Sep 2009 18:52:10 +0000 (+0000)
Subject:  Added the implementation section
X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=7e252eea8d7db4700b76c1734f467001b1b2a973;p=dealii-svn.git

 Added the implementation section

git-svn-id: https://svn.dealii.org/trunk@19460 0785d39b-7218-0410-832d-ea1e28bc413d
---

diff --git a/deal.II/examples/step-35/doc/intro.dox b/deal.II/examples/step-35/doc/intro.dox
index 1578b37346..58c209f946 100644
--- a/deal.II/examples/step-35/doc/intro.dox
+++ b/deal.II/examples/step-35/doc/intro.dox
@@ -16,7 +16,22 @@ Navier-Stokes equations. These equations describe the flow of a viscous incompre
   u_t + u \cdot \nabla u - \nu \Delta u + \nabla p = f, \\
   \nabla \cdot u = 0,
 @f}
-supplemented by the boundary condition $u=0$ Here $u$ represents the velocity of the flow and $p$ the pressure.
+where $u$ represents the velocity of the flow and $p$ the pressure. This system of equations is supplemented by
+the initial condition
+@f[
+  u |_{t=0} = u_0,
+@f]
+with $u_0$ sufficiently smooth and solenoidal, and suitable boundary conditions. For instance, an admissible boundary
+condition, is
+@f[
+  u|_{\partial\Omega} = u_b.
+@f]
+It is possible to prescribe other boundary conditions as well. In the test case that we solve here the boundary
+is partitioned into two disjoint subsets $\partial\Omega = \Gamma_1 \cup \Gamma_2$ and we have
+@f[
+  u|_{\Gamma_1} = u_b, \qquad u\times n|_{\Gamma_2} = 0.
+@f]
+where $n$ is the outer unit normal. The boundary condition on $\Gamma_2$ is used to model outflow conditions.
 
 In previous tutorial programs (see for instance @ref step_20 "step-20" and
 @ref step_22 "step-22") we have seen
@@ -106,7 +121,7 @@ projection methods that we have just described:
   @f]
   where
   @f[
-    H = \left\{ v \in L^2(\Omega)^d:\  \nabla\cdot v =0 \  v\cdot n|_{\partial\Omega} = 0 \right\}.
+    H = \left\{ v \in L^2(\Omega)^d:\  \nabla\cdot v =0, \  v\cdot n|_{\partial\Omega} = 0 \right\}.
   @f]
   Indeed, if we use this decomposition on $u^{k+1}$ we obtain
   @f[
@@ -114,13 +129,26 @@ projection methods that we have just described:
   @f]
   with $v^{k+1}\in H$. Taking the divergence of this equation we arrive at the projection equation.
   <li> The more accurate of the two variants outlined above is the rotational
-  one. That said, in the program below we use the standard form because it is
-  much easier to implement. Moreover, in the author's experience, it is the one that should be
-  used if, for instance, the viscosity $\nu$ is variable.
+  one. However, the program below implements both variants. Moreover, in the author's experience,
+  the standard form is the one that should be used if, for instance, the viscosity $\nu$ is variable.
 </ul>
 
+<a name = "implementation"></a>
+<h3> Implementation </h3>
+
+Our implementation of the projection methods follows <i>verbatim</i> the description given above. We must note,
+however, that as opposed to most other problems that have several solution components, we do not use
+vector-valued finite elements. Instead, we use separate finite elements for each component of the velocity
+and the pressure, respectively, and use different <code>DoFHandler</code>'s for those as well. The main
+reason for doing this is that, as we see from the description of the scheme, the <code>dim</code> components
+of the velocity and the pressure are decoupled. As a consequence, the equations for all the velocity components
+look all the same, have the same system matrix, and can be solved in parallel. Obviously, this approach
+has also its disadvantages. For instance, we need to keep several <code>DoFHandler</code>s and iterators
+synchronized when assembling matrices and right hand sides; obtaining quantities that are inherent to
+vector-valued functions (i.e. divergences) becomes a little awkward, and others.
+
 <a name ="testcase"></a>
-<h3> The testcase </h3>
+<h3> The Testcase </h3>
 
 The testcase that we use for this program consists of the flow around a square obstacle. The geometry is
 as follows: