From 980c39274919e27918c860dc41826d142b78a19b Mon Sep 17 00:00:00 2001 From: bonito Date: Tue, 4 Jan 2011 03:05:32 +0000 Subject: [PATCH] some more corrections, not done yet git-svn-id: https://svn.dealii.org/trunk@23118 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-38/doc/intro.dox | 8 ++++++-- 1 file changed, 6 insertions(+), 2 deletions(-) diff --git a/deal.II/examples/step-38/doc/intro.dox b/deal.II/examples/step-38/doc/intro.dox index a0e024ed85..abfb18ec63 100644 --- a/deal.II/examples/step-38/doc/intro.dox +++ b/deal.II/examples/step-38/doc/intro.dox @@ -45,12 +45,16 @@ 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 +Note that an alternate way to compute the surface gradient on smooth surfaces $\Gamma$ is @f[ -\nabla_S v := \nabla \tilde v - \mathbf n (\mathbf n \nabla \tilde v), +\nabla_S v = \nabla \tilde v - \mathbf n (\mathbf n \cdot \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 $\Gamma$. +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). +@f] 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 -- 2.39.5