From: Wolfgang Bangerth Date: Tue, 5 Jan 2021 17:19:33 +0000 (-0700) Subject: Clarify what the picture shows in step-74. X-Git-Tag: v9.3.0-rc1~661^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=0845a638e490efb2507236f4deb4adfeedd9653d;p=dealii.git Clarify what the picture shows in step-74. --- diff --git a/examples/step-74/doc/results.dox b/examples/step-74/doc/results.dox index e2bc069733..8d7590b48c 100644 --- a/examples/step-74/doc/results.dox +++ b/examples/step-74/doc/results.dox @@ -129,14 +129,18 @@ Cycle 2 @endcode The following figure provides a log-log plot of the errors versus -the number of degrees of freedom. Let $n$ be the number of degrees of +the number of degrees of freedom for this test case on the L-shaped +domain. In order to interpret it, let $n$ be the number of degrees of freedom, then on uniformly refined meshes, $h$ is of order $1/\sqrt{n}$ in 2D. Combining the theoretical results in the previous case, we see that if the solution is sufficiently smooth, we can expect the error in the $L_2$ norm to be of order $O(n^{-\frac{p+1}{2}})$ -and in $H^1$ seminorm to be $O(n^{-\frac{p}{2}})$. From the figure, we see -that the SIPG with adaptive mesh refinement produces desirable results -that match theoretical ones: +and in $H^1$ seminorm to be $O(n^{-\frac{p}{2}})$. It is not a priori +clear that one would get the same kind of behavior as a function of +$n$ on adaptively refined meshes like the ones we use for this second +test case, but one can certainly hope. Indeed, from the figure, we see +that the SIPG with adaptive mesh refinement produces asymptotically +the kinds of hoped-for results: