Reliability of a posteriori estimate techniques in finite element schemes
by
Carsten Carstensen
University of Kiel, Germany
Coauthors: Soeren Bartels (University of Kiel), Stefan A. Funken (University of Kiel)

The weak interpolation operator owing to Clément is one of the main mathematical tools in the residual based a posteriori finite element error analysis. We introduce an approximation interpolation operator [] to overcome assumptions on the regularity of the exact solution or on a saturation property of the numerical scheme in the a posteriori error analysis of finite element methods.

In our talk we will discuss a posteriori error estimates for the finite element computation for second order elliptic partial differential equations based on weighted residuals, local interface problems and averaging techniques. Emphasis is on reliability and efficiency constants [], reliability of ZZ-estimator on unstructered grids [], [], [] and a ranking of the schemes with theoretical and numerical evidence [].

Numerical examples are mentioned for the conforming, nonconforming, and mixed finite element schemes with applications to the Laplace, Stokes and locking-free Lamé problems that indicate the superiority of the averaging techniques.

References

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C. Carstensen, Weighted Clément-type interpolation an a posteriori analysis for fem. Berichtsreihe des Mathematischen Seminars Kiel, Technical report 97-19 Universität Kiel (1997). M²AN accepted for publication. (http://www.numerik.uni-kiel.de/reports/1997/97-19.html).

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C. Carstensen, S. Bartels, Each Averaging Technique Yields Reliable A Posteriori Error Control in FEM on Unstructured Grids Part I: Low Order Conforming, Nonconforming and Mixed FEM. In preparation (1999).

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C. Carstensen, S. Bartels, Each Averaging Technique Yields Reliable A Posteriori Error Control in FEM on Unstructured Grids Part II: Higher Order FEM. In preparation (1999).

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C. Carstensen, S. A. Funken, Fully reliable localised error control in the fem. Berichtsreihe des Mathematischen Seminars Kiel, Technical report 97-12 Universität Kiel (1997). SIAM J. Sci. Comp. accepted for publication. (http://www.numerik.uni-kiel.de/reports/1997/97-12.html).

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C. Carstensen, S. A. Funken, A posteriori error control in low-order finite element discretisations of incompressible stationary flow problems. Berichtsreihe des Mathematischen Seminars Kiel, Technical report 99-5 Universität Kiel (1999). (http://www.numerik.uni-kiel.de/reports/1999/99-5.html).

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C. Carstensen, S. A. Funken, Averaging technique for fe-a posteriori error control in elasticity. Part I: conforming FEM In preparation.

http://www.numerik.uni-kiel.de/cc

Date received: July 23, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadk-43.