The conditioning of boundary element equations on locally refined meshes and preconditioning by diagonal scaling

Consider a boundary integral operator on a bounded, d-dimensional surface in Rd+1. Suppose that the operator is a pseudodifferential operator of order 2m, m is an element of R, and that the associated bilinear form is symmetric and positive-definite. (The surface may be open or closed, and m may be positive or negative.) Let B denote the stiffness matrix arising from a Galerkin boundary element method with standard nodal basis functions. If local mesh refinement is used, then the partition may contain elements of very widely differing sizes, and consequently B may be very badly conditioned. In fact, if the elements are nondegenerate and 2\m\ < d then the l(2) condition number of B satisfies cond(B) less than or equal to CN2\m\/d (h(max)/h(min))(d-2m), where h(max) and h(min) are the sizes of the largest and smallest elements in the partition, and N is the number of degrees of freedom. However, if B is preconditioned using a simple diagonal scaling, then the condition number is reduced to O(N-2\m\/d). That is, diagonal scaling restores the condition number of the linear system to the same order of growth as that for a uniform partition. The growth in the critical case 2\m\ = d is worse by only a logarithmic factor.