From: kanschat Date: Thu, 24 Jun 2010 15:11:25 +0000 (+0000) Subject: add formulas X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=c5c8d14f8fed9e85522a3a350db92dad6cb9bb74;p=dealii-svn.git add formulas git-svn-id: https://svn.dealii.org/trunk@21328 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-39/doc/intro.dox b/deal.II/examples/step-39/doc/intro.dox index 3f224f0e33..2c51ebc953 100644 --- a/deal.II/examples/step-39/doc/intro.dox +++ b/deal.II/examples/step-39/doc/intro.dox @@ -8,4 +8,29 @@ residuals. All operators are implemented using the MeshWorker interface. Like in step-12, the discretization relies on finite element spaces, which are polynomial inside the mesh cells, but have no continuity -between cells. For such functions, we define the mean value and jump operators +between cells. Since such functions have two values on each face, one +from each side, we define mean value and jump operators as follows: +let K1 and K2 be the two cells +sharing a face, and let the traces of functions ui +and the outer normal vectors ni be labeled +accordingly. Then, on the face, we let +@f[ + \{\!\{ u \}\!\} = \frac{u_1 + u_2}2 +@f] + +Note, that if such an expression contains a normal vector, the +averaging operator turns into a jump. The interior penalty method for the problem +@f[ + -\Delta u = f \;\;\;\; u|_{\partial \Omega} = u^D +@f] +becomes +@f[ + \sum_{K\in T_h} (\nabla u, \nabla v)_K + \\ + + \sum_{F \in F_h^i} \biggl\{4\sigma_F (\{\!\{ u \mathbf n\}\!\}, \{\!\{ v \mathbf n \}\!\})_F + - 2 (\{\!\{ \nabla u \}\!\},\{\!\{ v\mathbf n \}\!\})_F + - 2 (\{\!\{ \nabla v \}\!\},\{\!\{ u\mathbf n \}\!\})_F + \biggr\} + \\ + = ??? +@f]