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Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM

Authors: Carsten Carstensen and Sören Bartels
Journal: Math. Comp. 71 (2002), 945-969
MSC (2000): Primary 65N30, 65R20, 74B20, 74G99, 74H99
Published electronically: February 4, 2002
MathSciNet review: 1898741
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Abstract: Averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing. In this paper, their reliablility is shown for conforming, nonconforming, and mixed low order finite element methods in a model situation: the Laplace equation with mixed boundary conditions. Emphasis is on possibly unstructured grids, nonsmoothness of exact solutions, and a wide class of averaging techniques. Theoretical and numerical evidence supports that the reliability is up to the smoothness of given right-hand sides.

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Additional Information

Carsten Carstensen
Affiliation: Institute for Applied Mathematics and Numerical Analysis, Vienna University of Technology, Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria

Sören Bartels
Affiliation: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel Ludewig-Meyn-Str. 4, D-24098 Kiel, FRG.

Keywords: A~posteriori error estimates, residual based error estimate, adaptive algorithm, reliability, finite element method, mixed finite element method, nonconforming finite element method
Received by editor(s): August 25, 1999
Published electronically: February 4, 2002
Article copyright: © Copyright 2002 American Mathematical Society