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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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Multilevel additive Schwarz method for the $h$-$p$ version of the Galerkin boundary element method
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by Norbert Heuer, Ernst P. Stephan and Thanh Tran PDF
Math. Comp. 67 (1998), 501-518 Request permission


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
  • Email:
  • Ernst P. Stephan
  • Affiliation: Institut für Angewandte Mathematik, Universität Hannover, Welfengarten 1, 30167 Hannover, Germany
  • Email:
  • Thanh Tran
  • Affiliation: School of Mathematics, The University of New South Wales, Sydney 2052, Australia
  • Email:
  • 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.
  • © Copyright 1998 American Mathematical Society
  • Journal: Math. Comp. 67 (1998), 501-518
  • MSC (1991): Primary 65N55, 65N38
  • DOI:
  • MathSciNet review: 1451325