From 0ad83f80f0cf8306f447949214fd2dad59ca45ea Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth <bangerth@colostate.edu> Date: Mon, 30 Apr 2018 22:26:14 -0600 Subject: [PATCH] Minor edits to the step-51 introduction. Specifically, make sure 'FE_DGPMonomial' is spelled correctly. While there, also address a couple of other things. --- examples/step-51/doc/intro.dox | 17 ++++++++++++----- 1 file changed, 12 insertions(+), 5 deletions(-) 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 <i>for each of the adjacent elements</i>, rather than just one, and similarly for edges and faces. As an example of how fast the number of -unknowns grows, -consider the <code>FE_DGP_Monomial</code> 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: <ol> <li> 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} -- 2.39.5