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 ℝ^d+1. Suppose that the operator is a pseudodifferential operator of order 2ℳ, ℳ ∈ 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 ℓ₂ condition number of B satisfies cond(B) < CN^2|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 script O sign(N²lℳl/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.