Prof. Carsten Carstensen,
HU Berlin
Averaging(FEM)4Everything?
Abstract:
The striking simplicity of averaging techniques and their amazing
accuracy in
too many numerical examples made them an extremly popular tool in
scientific
computing whenever finite elements might be useful. Given a dicrete flux
and an easily post-processed approximation
to
compute the error estimator
.
One does not even need
an equation to emply that technique occasionally named after
Zienkiewicz 
Zhu.
The beginning of a mathematical justification of the error estimator
as a computable
approximation of the (unknown) error
involved the
concept of super-convergence points. For highly structured meshes and a
very smooth exact solution 
, the error
of the
post-processed approximation
may be (much) smaller than
of the given .
Under the assumption that
is relatively sufficiently small,
the triangle inequality immediately verifies reliability, i.e.,
and efficiency, i.e.,
of the averaging error estimator
. However, the underlying
assumptions essentially contradict the notion of adaptive grid refining
for optimal experimental convergence rates when is singular. Moreover,
the proper treatment of boundary conditions lacks a serious inside.
The presentation reports on old and new arguments
for reliability and efficiency in the above sense with
multiplicative constants 
and 
and higher order terms
Hi-lighted are the general class of meshes, averaging
techniques, or finite element methods (conforming, nonconforming, and
mixed elements) for elliptic PDEs. Numerical examples illustrate the amazing
accuracy of
. The presentation closes with a
discussion on current developments and the limitations as well as the perspectives of averaging techniques.
C. Carstensen: Some remarks on the history and future of averaging techniques in finite element error analysis ZAMM 84 (2004) no. 1, 3-21.
| Zeit: | Freitag, 20. Februar 2004, 14.00 Uhr (Kaffee/Tee um 15.30) |
| Ort: | FU Berlin, Arnimallee 2-6, Raum 032 im EG |