From: Wolfgang Bangerth Date: Tue, 7 Sep 2010 12:55:22 +0000 (+0000) Subject: Add some text. X-Git-Tag: v8.0.0~5588 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=0663696a927bcca326d64be73b6090d2ebbc3e83;p=dealii.git Add some text. git-svn-id: https://svn.dealii.org/trunk@21864 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-12/doc/results.dox b/deal.II/examples/step-12/doc/results.dox index 738f56ecd3..02998aae3f 100644 --- a/deal.II/examples/step-12/doc/results.dox +++ b/deal.II/examples/step-12/doc/results.dox @@ -70,3 +70,30 @@ And finally we show a plot of a 3d computation. @image html step-12.sol-5-3d.png + + + +

Possibilities for extensions

+ +Given that the exact solution is known in this case, one interesting +avenue for further extensions would be to confirm the order of +convergence for this program. In the current case, the solution is +non-smooth, and so we can not expect to get a particularly high order +of convergence, even if we used higher order elements. But even if the +solution is smooth, the equation is not elliptic and so it is not +immediately clear that we should obtain a convergence order that +equals that of the optimal interpolation estimates (i.e. for example +that we would get $h^3$ convergence in the $L^2$ norm by using +quadratic elements). + +In fact, for hyperbolic equations, theoretical predictions often +indicate that the best one can hope for is an order one half below the +interpolation estimate. For example, for the streamline diffusion +method (an alternative method to the DG method used here to stabilize +the solution of the transport equation), one can prove that for +elements of degree $p$, the order of convergence is $p+\frac 12$ on +arbitrary meshes. While the observed order is frequently $p+1$ on +uniformly refined meshes, one can construct so-called Peterson meshes +on which the worse theoretical bound is actually attained. This should +be relatively simple to verify, for example using the +VectorTools::integrate_difference function.