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Solving Negative Order Equations by the Multigrid Method Via Variable Substitution

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Abstract

Variable substitutions are introduced to the single layer potential equations such that the order of pseudo-differential operator is changed from minus one to plus one. Though the condition number remains the same order after such a variable substitution, the frequencies of higher and lower eigenfunctions are switched. The multigrid iteration is shown to be an optimal order solver for the resulting linear systems of boundary element equations. Two types of variable substitutions are suggested. Numerical tests are presented showing efficiency of both methods, and supporting the theory.

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Acknowledgments

Liwei Xu is partially supported by the Youth 100 Plan start-up grant of Chongqing University (No. 0208001104413) in China.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

    George C. Hsiao & Shangyou Zhang

  2. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, People’s Republic of China

    Liwei Xu

  3. Institute of Scientific and Engineering Computing, Chongqing University, Chongqing, 400044, People’s Republic of China

    Liwei Xu

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  1. George C. Hsiao
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  2. Liwei Xu
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  3. Shangyou Zhang
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Correspondence to Shangyou Zhang.

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Hsiao, G.C., Xu, L. & Zhang, S. Solving Negative Order Equations by the Multigrid Method Via Variable Substitution. J Sci Comput 59, 371–385 (2014). https://doi.org/10.1007/s10915-013-9762-4

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  • DOI: https://doi.org/10.1007/s10915-013-9762-4

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