From bcab4507a695c5d097e0d40d931b1cea737b9cdc Mon Sep 17 00:00:00 2001 From: Scott Miller Date: Thu, 8 Aug 2013 14:09:27 +0000 Subject: [PATCH] Updates to HDG documentation git-svn-id: https://svn.dealii.org/trunk@30256 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-51/doc/intro.dox | 51 +++++++++++++++++++++++++- 1 file changed, 50 insertions(+), 1 deletion(-) diff --git a/deal.II/examples/step-51/doc/intro.dox b/deal.II/examples/step-51/doc/intro.dox index a5458f74da..784b5f7226 100644 --- a/deal.II/examples/step-51/doc/intro.dox +++ b/deal.II/examples/step-51/doc/intro.dox @@ -2,4 +2,53 @@

Introduction

This tutorial program presents the implementation of a hybridizable -discontinuous Galkerin method for the convection-diffusion equation. \ No newline at end of file +discontinuous Galkerin method for the convection-diffusion equation. + +

Hybridizable discontinuous Galerkin methods

+ +One common argument against the use of discontinuous Galerkin elements +is the large number of globally coupled degrees of freedom that one +must solve in an implicit system. For the FE_DGP_Monomial basis, each +scalar solution component is represented by polynomials of degree p +which yields (1/dim!)*\prod_{i=1}^{dim}(k+i) degrees of freedom per +element. Typically, all degrees of freedom in an element are coupled +to all of the degrees of freedom in the adjacent elements. The resulting +discrete equations yield very large linear systems very quickly, especially +for systems of equations in dim=2 or dim=3. + +To alleviate the computational cost of solving such large linear systems, +the hybridizable discontinuous Galerkin (HDG) methodology has recently been +developed by Cockburn and co-workers \cite{...}. The HDG method achieves +this goal by formulating the mathematical problem in such a way that the only +global unknowns are single-valued ``trace''-values on the skeleton of the +mesh, i.e. element edges. The DG solution on element interiors is thus decoupled +from the neighboring elements, enabling an element-by-element solution process +for the `interior' solution fields. + +- Discuss theoretical convergence rates. Other big picture HDG details? + +

HDG applied to the convection-diffusion problem

+ +- Problem formulation. Flux-formulation, HDG formulation, +domain, BCs, etc. + +- Reference: +@article{Nguyen20093232, +title = "An implicit high-order hybridizable discontinuous Galerkin method for linear convection–diffusion equations ", +journal = "Journal of Computational Physics ", +volume = "228", +number = "9", +pages = "3232 - 3254", +year = "2009", +note = "", +issn = "0021-9991", +doi = "http://dx.doi.org/10.1016/j.jcp.2009.01.030", +url = "http://www.sciencedirect.com/science/article/pii/S0021999109000308", +author = "N.C. Nguyen and J. Peraire and B. Cockburn", +keywords = "Finite element methods", +keywords = "Discontinuous Galerkin methods", +keywords = "Hybrid/mixed methods", +keywords = "Convection–diffusion equations " +} + +

Implementation

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