Spectral equivalence and proper clusters for matrices from the boundary element method


*Corresponding author for this work

Research output: Journal PublicationsJournal Article (refereed)peer-review

3 Citations (Scopus)


The Galerkin matrices An from applications of the boundary element method to integral equations of the first kind usually need to be preconditioned. In the Laplace equation context, we highlight a family of preconditioners Cn that simultaneously enjoy two important properties: (a) An and Cn are spectrally equivalent, and (b) the eigenvalues of Cn-1An have a proper cluster at unity. In the Helmholtz equation context, we prove the spectral equivalence for the so-called second Galerkin matrices and that the eigenvalues of Cn-1An still have a proper cluster at unity. We then show that some circulant integral approximate operator (CIAO) preconditioners belong to this family, including the well-known optimal CIAO. Consequently, if we use the preconditioned conjugate gradients to solve the problems, the number of iterations for a prescribed accuracy does not depend on n, and, what is more, the convergence rate is superlinear.

Original languageEnglish
Pages (from-to)1211-1224
Number of pages14
JournalInternational Journal for Numerical Methods in Engineering
Issue number9
Publication statusPublished - 30 Nov 2000
Externally publishedYes


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