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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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The analysis of multigrid algorithms for pseudodifferential operators of order minus one
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by James H. Bramble, Zbigniew Leyk and Joseph E. Pasciak PDF
Math. Comp. 63 (1994), 461-478 Request permission

Abstract:

Multigrid algorithms are developed to solve the discrete systems approximating the solutions of operator equations involving pseudodifferential operators of order minus one. Classical multigrid theory deals with the case of differential operators of positive order. The pseudodifferential operator gives rise to a coercive form on ${H^{ - 1/2}}(\Omega )$. Effective multigrid algorithms are developed for this problem. These algorithms are novel in that they use the inner product on ${H^{ - 1}}(\Omega )$ as a base inner product for the multigrid development. We show that the resulting rate of iterative convergence can, at worst, depend linearly on the number of levels in these novel multigrid algorithms. In addition, it is shown that the convergence rate is independent of the number of levels (and unknowns) in the case of a pseudodifferential operator defined by a single-layer potential. Finally, the results of numerical experiments illustrating the theory are presented.
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Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Math. Comp. 63 (1994), 461-478
  • MSC: Primary 65N55
  • DOI: https://doi.org/10.1090/S0025-5718-1994-1254145-2
  • MathSciNet review: 1254145