Summary
In some applications, the accuracy of the numerical solution of an elliptic problem needs to be increased only in certain parts of the domain. In this paper, local refinement is introduced for an overlapping additive Schwarz algorithm for the p-version finite element method. Both uniform and variable degree refinements are considered. The resulting algorithm is highly parallel and scalable.
In two and three dimensions, we prove an optimal bound for the condition number of the iteration operator under certain hypotheses on the refinement region. This bound is independent of the degree p, the number of subdomains N r and the mesh size H. In the general two dimensional case, we prove an almost optimal bound with polylogarithmic growth in p.
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This work was supported by the National Science Foundation under grant NSF-CCR-9120008, the U.S. Department of Energy under contract DE-FG-05-92ER25142 and the State of Texas under contract 1059. This work was also supported by the U.S. Department of Energy under contract DE-FG02-88ER25053 and by the National Science Foundation under Grant NSF-CCR-8903003, while the author was a student at the Courant Institute of Mathematical Science, 251 Mercer Street, New York, NY 10012
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Pavarino, L.F. Schwarz methods with local refinement for the p-version finite element method. Numer. Math. 69, 185–211 (1994). https://doi.org/10.1007/s002110050087
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DOI: https://doi.org/10.1007/s002110050087