From bc06cb8ff8dc4cf0f3bceae236e941cca76e9595 Mon Sep 17 00:00:00 2001 From: Martin Kronbichler Date: Mon, 16 Sep 2013 20:50:27 +0000 Subject: [PATCH] Add a comment on vector-valued problems git-svn-id: https://svn.dealii.org/trunk@30746 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-51/doc/intro.dox | 20 +++++++++++++------- 1 file changed, 13 insertions(+), 7 deletions(-) diff --git a/deal.II/examples/step-51/doc/intro.dox b/deal.II/examples/step-51/doc/intro.dox index dacb532c60..daa8de30fe 100644 --- a/deal.II/examples/step-51/doc/intro.dox +++ b/deal.II/examples/step-51/doc/intro.dox @@ -262,13 +262,19 @@ super-convergence result with rate $\mathcal {O}(h^{p+2})$. The motivation for the above equation is the minimization of the expression $|\kappa \nabla u_h^* + \mathbf{q}_h|^2$ over the cell K. It should be noted that there is some freedom in constructing $u_h^*$ and this approach to extract the -information from the gradient is not the only one. For example, the paper by -Nguyen, Peraire and Cockburn cited above suggests another somewhat more -involved formula for convection-diffusion that can also post-process the flux -variable into an $H(\Omega,\mathrm{div})$-conforming variant and better -represents the local convection-diffusion operator when the diffusion is -small. We leave the implementation of a more sophisticated post-processing as -a possible extension to the user. +information from the gradient is not the only one. In particular, the +post-processed solution defined here does not satisfy the convection-diffusion +equation in any sense. As an alternative, the paper by Nguyen, Peraire and +Cockburn cited above suggests another somewhat more involved formula for +convection-diffusion that can also post-process the flux variable into an +$H(\Omega,\mathrm{div})$-conforming variant and better represents the local +convection-diffusion operator when the diffusion is small. We leave the +implementation of a more sophisticated post-processing as a possible extension +to the interested reader. + +Note that for vector-valued problem, the approach is very similar. One simply +sets the constraint for the mean value of each vector component separately and +uses the gradient for the main work.

Problem specific data

-- 2.39.5