From 84960676922976a875d78f7ed9d9091d0faa0ac5 Mon Sep 17 00:00:00 2001 From: Andrea Bonito Date: Mon, 3 Jan 2011 22:41:07 +0000 Subject: [PATCH] step-38 some modifs git-svn-id: https://svn.dealii.org/trunk@23116 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-38/doc/intro.dox | 12 +++++++++--- 1 file changed, 9 insertions(+), 3 deletions(-) diff --git a/deal.II/examples/step-38/doc/intro.dox b/deal.II/examples/step-38/doc/intro.dox index 18cd414ad1..144040a99e 100644 --- a/deal.II/examples/step-38/doc/intro.dox +++ b/deal.II/examples/step-38/doc/intro.dox @@ -31,7 +31,7 @@ i.e. each point $\hat{\mathbf x}\in\hat S$ induces a point ${\mathbf @f[ G_S:= (D \mathbf{x}_S)^T \ D \mathbf{x}_S @f] -denote the corresponding first fundamental form, where $D +denotes the corresponding first fundamental form, where $D \mathbf{x}_S=\left(\frac{\partial x_{S,i}(\hat{\mathbf x})}{\partial \hat x_j}\right)_{ij}$ is the derivative (Jacobian) of the mapping. In the following, $S$ will be either the entire surface $\Gamma$ or, @@ -45,6 +45,12 @@ R$ by @f] The surface Laplacian (also called the Laplace-Beltrami operator) is then defined as $\Delta_S:= \nabla_S \cdot \nabla_S$. +Note that an alternate way to define the surface gradient on smooth surfaces $\Gamma$ is +@f[ +\nabla_S v := \nabla \tilde v - \mathbf n (\mathbf n \nabla \tilde v), +@f] +where $\tilde v$ is a "smooth" extension of $v$ in a tubular neighborhood of $\Gamma$ and +$\mathbf n$ is the normal of $\Gamma$. As usual, we are only interested in weak solutions for which we can use $C^0$ finite elements (rather than requiring $C^1$ continuity as for strong @@ -132,9 +138,9 @@ We produce one test case for a 2d problem and another one for 3d:
  • In 2d, let's choose as domain a half circle. On this domain, we choose the function $u(\mathbf x)=-2x_1x_2$ as the solution. To compute the right hand - side, we have to compute the second tangential derivatives of the + side, we have to compute the surface Laplacian of the solution function. There are (at least) two ways to do that. The first one - is to project away the normal derivative, i.e. to compute + 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 = -- 2.39.5