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An efficient computation of the inverse of the single layer matrix for the resolution of the linear elasticity problem in BEM

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Abstract

In this paper, we describe and analyze fast procedures for numerical resolution of Dirichlet and mixed boundary value problems in linear elasticity, in the particular case of two-dimensional circular domain. The direct boundary integral formulation of the Dirichlet problem is approximated by using the B-splines on the boundary curve. This yields an algebraic linear system involving two dense matrices with block structure, the single layer and the double layer matrices. The associated matrix entries are computed explicitly and efficiently. Additionally, the circulant block structure of the single layer matrix and the discrete Fourier transform (DFT) matrix enable us to easily compute the inverse of the single layer matrix as a product of sparse matrices and the discrete Fourier transform matrix which speed up the matrix-vector multiplication. Moreover, the discrete Fourier transform permits us to construct some optimal preconditioners for the conjugate gradient methods. Some numerical examples for the Dirichlet and the mixed problems which show a remarkable efficiency and accuracy of algorithms for the problems are presented.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Dschang, P.O. Box 67, Dschang, Cameroon

    Lionel Ouya Ndjansi & Laurent Tchoualag

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  1. Lionel Ouya Ndjansi
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  2. Laurent Tchoualag
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Correspondence to Lionel Ouya Ndjansi.

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Communicated by: Anna-Karin Tornberg

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Ndjansi, L.O., Tchoualag, L. An efficient computation of the inverse of the single layer matrix for the resolution of the linear elasticity problem in BEM. Adv Comput Math 49, 31 (2023). https://doi.org/10.1007/s10444-023-10039-x

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