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

DOI:
https://doi.org/10.1090/S0025-5718-98-00993-4

MathSciNet review:
1489969

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Abstract | References | Similar Articles | Additional Information

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 -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

Email:
willy@mathematik.uni-freiburg.de

**M. Rumpf**

Affiliation:
Institut für Angewandte Mathematik, Universität Bonn, Wegelerstrasse 6, D-52115 Bonn, Germany

Email:
rumpf@iam.uni-bonn.de

DOI:
https://doi.org/10.1090/S0025-5718-98-00993-4

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