From: bonito Date: Wed, 5 Jan 2011 20:35:23 +0000 (+0000) Subject: git-svn-id: https://svn.dealii.org/trunk@23127 0785d39b-7218-0410-832d-ea1e28bc413d X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=b13916ca2e9ea686cacf059e685b5a1f40be59bf;p=dealii-svn.git git-svn-id: https://svn.dealii.org/trunk@23127 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-38/doc/intro.dox b/deal.II/examples/step-38/doc/intro.dox index efac20be85..5a1d33c6e9 100644 --- a/deal.II/examples/step-38/doc/intro.dox +++ b/deal.II/examples/step-38/doc/intro.dox @@ -53,7 +53,7 @@ where $\tilde v$ is a "smooth" extension of $v$ in a tubular neighborhood of $\G $\mathbf n$ is the normal of $\Gamma$. Since $\Delta_S = \nabla_S \cdot \nabla_S$, we deduce @f[ -\Delta_S v = \Delta \tilde v - \mathbf n^T \ D \tilde v \ \mathbf n - (\nabla \tilde v)\cdot \mathbf n (\nabla \cdot \mathbf n - \mathbf n \ D \mathbf n \ \mathbf n ). +\Delta_S v = \Delta \tilde v - \mathbf n^T \ D^2 \tilde v \ \mathbf n - (\mathbf n \cdot \nabla \tilde v) (\nabla \cdot \mathbf n - \mathbf n^T \ D \mathbf n \ \mathbf n ). @f] Worth mentioning, the term $\nabla \cdot \mathbf n - \mathbf n \ D \mathbf n \ \mathbf n$ appearing in the above expression is the total curvature of the surface (sum of principal curvatures). @@ -148,7 +148,7 @@ We produce one test case for a 2d problem and another one for 3d: solution function. There are (at least) two ways to do that. The first one is to project away the normal derivative as described above using the natural extension of $u(\mathbf x)$ (still denoted by $u$) over $\mathbb R^d$, i.e. to compute @f[ - -\Delta_\Gamma u = \Delta u - \mathbf n^T \ D u \ \mathbf n - (\nabla u)\cdot \mathbf n \kappa, + -\Delta_\Gamma u = \Delta u - \mathbf n^T \ D^2 u \ \mathbf n - (\mathbf n \cdot \nabla u)\ \kappa, @f] where $\kappa$ is the total curvature of $\Gamma$. Since we are on the unit circle, $\mathbf n=\mathbf x$ and $\kappa = 1$ so that