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Multilevel additive Schwarz method for the $h$-$p$ version of the Galerkin boundary element method

Authors: Norbert Heuer, Ernst P. Stephan and Thanh Tran
Journal: Math. Comp. 67 (1998), 501-518
MSC (1991): Primary 65N55, 65N38
MathSciNet review: 1451325
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Abstract: We study a multilevel additive Schwarz method for the $h$-$p$ version of the Galerkin boundary element method with geometrically graded meshes. Both hypersingular and weakly singular integral equations of the first kind are considered. As it is well known the $h$-$p$ version with geometric meshes converges exponentially fast in the energy norm. However, the condition number of the Galerkin matrix in this case blows up exponentially in the number of unknowns $M$. We prove that the condition number $\kappa (P)$ of the multilevel additive Schwarz operator behaves like $O(\sqrt {M}\log ^2M)$. As a direct consequence of this we also give the results for the $2$-level preconditioner and also for the $h$-$p$ version with quasi-uniform meshes. Numerical results supporting our theory are presented.

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Additional Information

Norbert Heuer
Affiliation: Institut für Wissenschaftliche Datenverarbeitung, Universität Bremen, Postfach 330440, 28334 Bremen, Germany
MR Author ID: 314970

Ernst P. Stephan
Affiliation: Institut für Angewandte Mathematik, Universität Hannover, Welfengarten 1, 30167 Hannover, Germany

Thanh Tran
Affiliation: School of Mathematics, The University of New South Wales, Sydney 2052, Australia

Keywords: $h$-$p$ version boundary integral equation method, additive Schwarz operator, multilevel method, preconditioned conjugate gradient algorithm
Received by editor(s): October 24, 1995
Received by editor(s) in revised form: November 18, 1996
Additional Notes: This work was started while the third author was visiting the Institut für Angewandte Mathematik at the University of Hannover. The work was partly supported by the DFG research group “Zuverlässigkeit von Modellierung und Berechnung in der Angewandten Mechanik” at the University of Hannover.
Article copyright: © Copyright 1998 American Mathematical Society