New wavelet preconditioner for solving boundary integral equations over nonsmooth boundaries

Abstract

Matrices obtained by wavelet discretisation of partial differential equations or boundary integral equations (BIEs) are typically sparse with a finger-like sparsity pattern, in contrast to matrices obtained by traditional single scale discretisation, which are dense. In some cases diagonal preconditioning is sufficient, but there are effective preconditioners that can be used when this is not the case. Here, we consider the particular case of BIEs whose boundary has a geometrical singularity, and propose a new preconditioning strategy based on permutations of the unknowns. The strategy's performance is analysed and compared with related techniques for the double layer equation for Laplace's equation.

^*E-mail: [email protected]. Support of a UK EPSRC grant (GR/R22315) is gratefully acknowledged.

Keywords:

• Preconditioning
• Continuous wavelet discretisation
• Boundary integral equation
• Permutation
• Geometric singularity

Notes

^*E-mail: [email protected]. Support of a UK EPSRC grant (GR/R22315) is gratefully acknowledged.