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

The Galerkin matrices A_n 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 C_n that simultaneously enjoy two important properties: (a) A_n and C_n are spectrally equivalent, and (b) the eigenvalues of C_n^-1A_n 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_n^-1A_n 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.