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

The Galerkin matrices An from applications of the boundary element method to integral equations of the rst 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 equiv- alent, and (b) the eigenvalues of C−1 n An 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 C−1 n An 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.