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. Given a dicrete flux
and an easily post-processed approximation
to compute
the error estimator
.
One does not even need
an equation to employ 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
h.o.t.
is relatively sufficiently small,
the triangle inequality immediately verifies reliability, i.e.,
The presentation reports on old and new arguments
for reliability and efficiency in the above sense with
multiplicative constants and
and higher order terms
h.o.t. 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.
Future talks in Scientific Computing Seminar
Nov. 9: Jim Douglas, Jr., Department of Mathematics, Purdue University.
Nov. 11: Jiwen He, Department of Mathematics, University of Houston.
Nov. 18: E.W. Sachs, Department of Mathematics, Virginia Tech.
Nov. 22: P. Bochev: Sandia National Laboratories.
Nov. 23: O. Pironneau, Universite Pierre-et-Marie-Curie, France.
This seminar is easily accessible to persons with disabilities.
For more information or for assistance, please contact the Mathematics
Department at 743-3500.