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Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part II: Higher order FEM

Authors: Sören Bartels and Carsten Carstensen
Journal: Math. Comp. 71 (2002), 971-994
MSC (2000): Primary 65N30, 65R20, 74B20, 74G99, 74H99
Published electronically: February 4, 2002
MathSciNet review: 1898742
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Abstract: Averaging techniques are popular tools in adaptive finite element methods since they provide efficient a posteriori error estimates by a simple postprocessing. In the second paper of our analysis of their reliability, we consider conforming $h$-FEM of higher (i.e., not of lowest) order in two or three space dimensions. In this paper, reliablility is shown for conforming higher 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 local 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

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

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

Keywords: A posteriori error estimates, residual based error estimate, adaptive algorithm, reliability, finite element method, higher order finite element method
Received by editor(s): February 17, 2000
Published electronically: February 4, 2002
Article copyright: © Copyright 2002 American Mathematical Society