From: Scott Miller Date: Fri, 6 Sep 2013 11:13:25 +0000 (+0000) Subject: Minor intro update X-Git-Tag: v8.1.0~871 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=9297e59c51d5f5f7fe14174b0b20696bc72e058c;p=dealii.git Minor intro update git-svn-id: https://svn.dealii.org/trunk@30627 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 1c14fa37c9..20c8ed7f91 100644 --- a/deal.II/examples/step-51/doc/intro.dox +++ b/deal.II/examples/step-51/doc/intro.dox @@ -20,13 +20,11 @@ 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 -( - N.C. Nguyen and J. Peraire: +(N.C. Nguyen and J. Peraire: Hybridizable discontinuous Galerkin methods for partial differential equations in continuum mechanics, Journal of Computational Physics, 2012, 231:18, 5955-5988. - [DOI] -). + [DOI]). 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 @@ -35,13 +33,13 @@ point the single-valued ``trace'' values on the skeleton of the mesh, i.e. element edges, are taken to be independent unknown quantities. The Dirichlet-to-Neumann map concept then permits the solution procedure:
    -
  1. Solve a Dirichlet problem on each element individually, where the local/interior -element solutions no longer couple to neighboring elements. This is known as the +
  2. Use local element interior data to enforce a Neumann condition on the +skeleton of the triangulation. The global problem is then to solve for the +trace values, which are the only globally coupled unknowns. +
  3. Use the known skeleton values as Dirichlet data for solving local +element-level solutions. This is known as the 'local solver', and is an embarrassingly parallel element-by-element solution process. -
  4. Use the local element equations to construct the global problem for the -trace values. These are the only globally coupled unknowns. -
  5. Solve for the element-local solutions using the computed trace values.
The above procedure also has a linear algebra interpretation and is referred to @@ -79,9 +77,9 @@ problem) and subtract $CA^{-1}B$ from $D$.

Solution quality and rates of convergence

Another criticism of traditional DG methods is that the approximate fluxes converge suboptimally. The local HDG solutions can be shown to converge -as $\mathcal{O}(h^{p+1})$, i.e., at optimal order. Additionally, a super-convergence property can -be used to post-process a new approximate solution that converges at the -rate $\mathcal{O}(h^{p+2})$. +as $\mathcal{O}(h^{p+1})$, i.e., at optimal order. Additionally, a +super-convergence property can be used to post-process a new approximate +solution that converges at the rate $\mathcal{O}(h^{p+2})$. @@ -153,6 +151,7 @@ Eliminating $\hat{u}$ from the weak form in favor of $\lambda$, summing the elem contributions across all elements in the triangulation, and enforcing the normal component of the numerical flux, we arrive at the final form of the problem: Find $(\mathbf{q}, u, \lambda)$ such that +TODO: update this equation! @f{eqnarray*} (\kappa^{-1} \mathbf{q}, \mathbf{v})_{\mathcal{T}} - (u, \nabla\cdot\mathbf{v})_{\mathcal{T}} @@ -172,7 +171,6 @@ Find $(\mathbf{q}, u, \lambda)$ such that @f} -

Problem specific data

Example specific data: $\Omega := [-1,1]^d$