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An adaptive strategy for elliptic problems including a posteriori controlled boundary approximation

Authors: W. Dörfler and M. Rumpf
Journal: Math. Comp. 67 (1998), 1361-1382
MSC (1991): Primary 65N15, 65N30, 65N50
MathSciNet review: 1489969
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Abstract: We derive a posteriori error estimates for the approximation of linear elliptic problems on domains with piecewise smooth boundary. The numerical solution is assumed to be defined on a Finite Element mesh, whose boundary vertices are located on the boundary of the continuous problem. No assumption is made on a geometrically fitting shape. A posteriori error estimates are given in the energy norm and the $L^2$-norm, and efficiency of the adaptive algorithm is proved in the case of a saturated boundary approximation. Furthermore, a strategy is presented to compute the effect of the non-discretized part of the domain on the error starting from a coarse mesh. This especially implies that parts of the domain, where the measured error is small, stay non-discretized. The presented algorithm includes a stable path following to supply a sufficient polygonal approximation of the boundary, the reliable computation of the a posteriori estimates and a mesh adaptation based on Delaunay techniques. Numerical examples illustrate that errors outside the initial discretization will be detected.

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Additional Information

W. Dörfler
Affiliation: Institut für Angewandte Mathematik, Universität Freiburg, Hermann-Herder- Strasse 10, D-79104 Freiburg, Germany

M. Rumpf
Affiliation: Institut für Angewandte Mathematik, Universität Bonn, Wegelerstrasse 6, D-52115 Bonn, Germany
MR Author ID: 604100

Keywords: Adaptive mesh refinement, a posteriori error estimate, boundary approximation, Poisson’s equation
Received by editor(s): March 4, 1996
Received by editor(s) in revised form: January 23, 1997
Article copyright: © Copyright 1998 American Mathematical Society