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

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- by W. Dörfler and M. Rumpf PDF
- Math. Comp.
**67**(1998), 1361-1382 Request permission

## 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.## References

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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
- MR Author ID: 604100
- Email: rumpf@iam.uni-bonn.de
- Received by editor(s): March 4, 1996
- Received by editor(s) in revised form: January 23, 1997
- © Copyright 1998 American Mathematical Society
- 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