From: kronbichler Date: Wed, 18 Sep 2013 14:48:18 +0000 (+0000) Subject: Explain integrals X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=e312e6d3274b2480b70ac15749ce35ab32ba9b18;p=dealii-svn.git Explain integrals git-svn-id: https://svn.dealii.org/trunk@30795 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-51/doc/intro.dox b/deal.II/examples/step-51/doc/intro.dox index 0badf15706..56ead0eb28 100644 --- a/deal.II/examples/step-51/doc/intro.dox +++ b/deal.II/examples/step-51/doc/intro.dox @@ -211,6 +211,14 @@ The unknowns $(\mathbf{q}_h, u_h)$ are referred to as local variables; they are represented as standard DG variables. The unknown $\hat{u}_h$ is the skeleton variable which has support on the codimension-1 surfaces (faces) of the mesh. +We use the notation $(\cdot, \cdot)_{\mathcal{T}} = \sum_K (\cdot, \cdot)_K$ +to denote the sum of integrals over all cells and $\left<\cdot, +\cdot\right>_{\partial \mathcal{T}} = \sum_K \left<\cdot, +\cdot\right>_{\partial K}$ to denote integration over all faces of all cells, +i.e., interior faces are visited twice. When combining the contribution from +both elements sharing a face, the above equation yields terms familiar for DG +with jumps of the solution over the cell boundaries. + In the equation above, the space $\mathcal {W}_h^{p}$ for the scalar variable uh is defined as the space of functions that are tensor product polynomials of degree p on each cell and discontinuous over the