From 706b78ebda303e31c7c9dd410a23b746ba1505b6 Mon Sep 17 00:00:00 2001 From: wolf Date: Fri, 3 May 2002 12:58:27 +0000 Subject: [PATCH] Explain error estimates for derivative. git-svn-id: https://svn.dealii.org/trunk@5801 0785d39b-7218-0410-832d-ea1e28bc413d --- .../step-14.data/results.html | 19 +++++++++++++++++-- 1 file changed, 17 insertions(+), 2 deletions(-) diff --git a/deal.II/doc/tutorial/chapter-2.step-by-step/step-14.data/results.html b/deal.II/doc/tutorial/chapter-2.step-by-step/step-14.data/results.html index 68fbd40e42..758af0a481 100644 --- a/deal.II/doc/tutorial/chapter-2.step-by-step/step-14.data/results.html +++ b/deal.II/doc/tutorial/chapter-2.step-by-step/step-14.data/results.html @@ -385,12 +385,27 @@ see a comparison of true and estimated error: It is obvious that here the error estimates are not as good as -previously, under-estimation the error by about a factor of 10. At +previously, under-estimation the error by about a factor of 2-4. At least the sign is correct, leading to a slight improvement in the estimated values if we sum computed value and estimated error.

-TODO: explanation! +

+The difference between true and estimated error can be tracked down to +the bad approximation of the dual solution. After all, in 2d, the dual +solution has a 1/r singularity near the evaluation point +(while the dual solution for the point value only has a logarithmic +singularity there), which does not allow for a good approximation of +the dual solution by any finite element space. Indeed, computing the +dual solution with even higher order (i.e. cubic or quartic) finite +elements does not significantly improve the quality of error +estimates. Intuitively, the reason is that the numerical approximation +cannot follow accurately the simgularity of the dual solution; its +resulting values near the point of evaluation are thus too small, and +the error is underestimated there. Since the error is very much +concentrated near the point of evaluation, this spoils the entire +estimate. +

Step-13 revisited

-- 2.39.5