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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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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


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
  • Email:
  • M. Rumpf
  • Affiliation: Institut für Angewandte Mathematik, Universität Bonn, Wegelerstrasse 6, D-52115 Bonn, Germany
  • MR Author ID: 604100
  • Email:
  • 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:
  • MathSciNet review: 1489969