From: Wolfgang Bangerth Date: Tue, 1 May 2018 04:26:14 +0000 (-0600) Subject: Minor edits to the step-51 introduction. X-Git-Tag: v9.0.0-rc1~39^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=refs%2Fpull%2F6408%2Fhead;p=dealii.git Minor edits to the step-51 introduction. Specifically, make sure 'FE_DGPMonomial' is spelled correctly. While there, also address a couple of other things. --- diff --git a/examples/step-51/doc/intro.dox b/examples/step-51/doc/intro.dox index 9754c5170f..6a72e06e63 100644 --- a/examples/step-51/doc/intro.dox +++ b/examples/step-51/doc/intro.dox @@ -18,8 +18,7 @@ must solve in an implicit system. This is because, unlike continuous finite elements, in typical discontinuous elements there is one degree of freedom at each vertex for each of the adjacent elements, rather than just one, and similarly for edges and faces. As an example of how fast the number of -unknowns grows, -consider the FE_DGP_Monomial basis: each +unknowns grows, consider the FE_DGPMonomial basis: each scalar solution component is represented by polynomials of degree $p$ with $(1/dim!)*\prod_{i=1}^{dim}(p+i)$ degrees of freedom per element. Typically, all degrees of freedom in an element are coupled @@ -41,8 +40,15 @@ The HDG method achieves this goal by formulating the mathematical problem using Dirichlet-to-Neumann mappings. The partial differential equations are first written as a first order system, and each field is then discretized via a DG method. At this -point the single-valued "trace" values on the skeleton of the +point, the single-valued "trace" values on the skeleton of the mesh, i.e. element faces, are taken to be independent unknown quantities. +This yields unknowns in the discrete formulation that fall into two categories: +- Face unknowns that only couple with the unknowns from both sides of the face; +- Cell unknowns that only couple with each other and the other unknowns + defined within the same cell. Crucially, no cell interior degree of freedom + on one cell ever couples to any interior cell degree of freedom of a + different cell. + The Dirichlet-to-Neumann map concept then permits the following solution procedure:
  1. Use local element interior data to enforce a Neumann condition on the @@ -56,8 +62,9 @@ solution process. The above procedure also has a linear algebra interpretation and is referred to as static condensation. Let us write the complete linear system associated to -the HDG problem as a block system with the discrete DG variables $U$ as -first block and the skeleton variables $\Lambda$ as the second block: +the HDG problem as a block system with the discrete DG (cell interior) +variables $U$ as first block and the skeleton (face) variables $\Lambda$ as the +second block: @f{eqnarray*} \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} U \\ \Lambda \end{pmatrix}