The Institute for Theoretical and Engineering Science
Department of Mathematics, University of Houston

Scientific Computing Seminars

Professor Carsten Carstensen
Department of Mathematics
Humboldt-Universität zu Berlin, Germany

Averaging techniques for
a posteriori finite element error control

Tuesday, November 2$^*$
2:30 PM- 3:30 PM
Room 634 S&R1

$^*$Note: Special Date and Time

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 $p_h$ and an easily post-processed approximation $\mathcal{A}p_h$ to compute the error estimator $\eta_{\mathcal{A}}:=\Vert p_h-\mathcal{A}p_h\Vert$. 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 $\eta_{\mathcal{A}}$ as a computable approximation of the (unknown) error $\Vert p-p_h\Vert$ involved the concept of super-convergence points. For highly structured meshes and a very smooth exact solution $p$, the error $\Vert p-\mathcal{A}p_h\Vert$ of the post-processed approximation $\mathcal{A}p_h$ may be (much) smaller than $\Vert p-p_h\Vert$ of the given $p_h$. Under the assumption that $\Vert p-\mathcal{A}p_h\Vert=$ h.o.t. is relatively sufficiently small, the triangle inequality immediately verifies reliability, i.e.,

$$\Vert p-p_h\Vert\le C_{rel} \eta_{\mathcal{A}}+$$ h.o.t.$$,$$

and efficiency, i.e.,

$$\eta_{\mathcal{A}}\le C_{eff} \Vert p-p_h\Vert +$$ h.o.t.$$,$$

of the averaging error estimator $\eta_{\mathcal{A}}$. However, the underlying assumptions essentially contradict the notion of adaptive grid refining for optimal experimental convergence rates when $p$ 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 $C_{rel}$ and $C_{eff}$ 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 $\eta_{\mathcal{A}}$. 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

$\bullet$ Nov. 9: Jim Douglas, Jr., Department of Mathematics, Purdue University.

$\bullet$ Nov. 11: Jiwen He, Department of Mathematics, University of Houston.

$\bullet$ Nov. 18: E.W. Sachs, Department of Mathematics, Virginia Tech.

$\bullet$ Nov. 22: P. Bochev: Sandia National Laboratories.

$\bullet$ Nov. 23: O. Pironneau, Universite Pierre-et-Marie-Curie, France.

$\bullet$ This seminar is easily accessible to persons with disabilities. For more information or for assistance, please contact the Mathematics Department at 743-3500.