Analytical preconditioners for Neumann elastodynamic boundary element methods

Stéphanie Chaillat (), Marion Darbas () and Frédérique Le Louër ()
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Stéphanie Chaillat: Institut Polytechnique de Paris
Marion Darbas: Institut Galilée, Université Sorbonne Paris Nord
Frédérique Le Louër: LMAC EA 2222, Sorbonne Universités, Université de technologie de Compiègne

Partial Differential Equations and Applications, 2021, vol. 2, issue 2, 1-26

Abstract: Abstract Recent works in the boundary element method (BEM) community have been devoted to the derivation of fast techniques to perform the matrix-vector product needed in the iterative solver. Fast BEMs are now very mature. However, it has been shown that the number of iterations can significantly hinder the overall efficiency of fast BEMs. The derivation of robust preconditioners is now inevitable to increase the size of the problems that can be considered. Analytical preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead of discretization. The main contribution of this paper is to propose new analytical preconditioners to treat Neumann exterior scattering problems in 2D and 3D elasticity. These preconditioners are local approximations of the adjoint Neumann-to-Dirichlet map. We propose three approximations with different orders. The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). An analytical spectral study confirms the expected behavior of the preconditioners, i.e., a better eigenvalue clustering especially in the elliptic part contrary to the standard CFIE of the first-kind. We provide various 2D numerical illustrations of the efficiency of the method for smooth and non smooth geometries. In particular, the number of iterations is shown to be independent of the density of discretization points per wavelength which is not the case for the standard CFIE. In addition, it is less sensitive to the frequency.

Keywords: Scattering; Time-harmonic elastic waves; Boundary element method; Analytical preconditioner; Approximate local Neumann-to-Dirichlet map; Cavity; 65N38; 65F08; 45B05; 47G30 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s42985-021-00075-x

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