From 706b78ebda303e31c7c9dd410a23b746ba1505b6 Mon Sep 17 00:00:00 2001
From: wolf <wolf@0785d39b-7218-0410-832d-ea1e28bc413d>
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:
   </tr>
 </table>
 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.
 </p>
 
-TODO: explanation!
+<p>
+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 <em>1/r</em> 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.
+</p>
 
 
 <h2>Step-13 revisited</h2>
-- 
2.39.5