From: Timo Heister Date: Wed, 6 Nov 2019 15:41:43 +0000 (-0500) Subject: remove extension ideas from step-12b X-Git-Tag: v9.2.0-rc1~836^2~1 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=603d5ead9685e5d0630fa1791ff1cd1e45596418;p=dealii.git remove extension ideas from step-12b --- diff --git a/examples/step-12b/doc/results.dox b/examples/step-12b/doc/results.dox index 22e824300f..ee39c3ffe5 100644 --- a/examples/step-12b/doc/results.dox +++ b/examples/step-12b/doc/results.dox @@ -26,113 +26,7 @@ And finally we show a plot of a 3d computation. - -

Why use discontinuous elements

- -In this program we have used discontinuous elements. It is a legitimate -question to ask why not simply use the normal, continuous ones. Of course, to -everyone with a background in numerical methods, the answer is obvious: the -continuous Galerkin (cG) method is not stable for the transport equation, -unless one specifically adds stabilization terms. The DG method, however, -is stable. Illustrating this with the current program is not very -difficult; in fact, only the following minor modifications are necessary: -- Change the element to FE_Q instead of FE_DGQ. -- Add handling of hanging node constraints in exactly the same way as step-6. -- We need a different solver; the direct solver in step-29 is a convenient - choice. -An experienced deal.II user will be able to do this in less than 10 minutes. - -While the 2d solution has been shown above, containing a number of small -spikes at the interface that are, however, stable in height under mesh -refinement, results look much different when using a continuous element: - - - - - - - - - - - - - - - - - - - - -
- 0   - - - - 1   - - -
- 2   - - - - 3   - - -
- 4   - - - - 5   - - -
- -In refinement iteration 5, the image can't be plotted in a reasonable way any -more as a 3d plot. We thus show a color plot with a range of $[-1,2]$ (the -solution values of the exact solution lie in $[0,1]$, of course). In any case, -it is clear that the continuous Galerkin solution exhibits oscillatory -behavior that gets worse and worse as the mesh is refined more and more. - -There are a number of strategies to stabilize the cG method, if one wants to -use continuous elements for some reason. Discussing these methods is beyond -the scope of this tutorial program; an interested reader could, for example, -take a look at step-31. - - -

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. - -A different direction is to observe that the solution of transport problems -often has discontinuities and that therefore a mesh in which we bisect -every cell in every coordinate direction may not be optimal. Rather, a better -strategy would be to only cut cells in the direction parallel to the -discontinuity. This is called anisotropic mesh refinement and is the -subject of step-30. +For ideas for further extensions, please see see step-12.