The Conditioning of Boundary Element Equations on Locally Refined Meshes and Preconditioning by Diagonal Scaling

Abstract

Consider a boundary integral operator on a bounded,d-dimensional surface in $\mbox{\smallBbb R}^{d+1}$. Suppose that the operator is a pseudodifferential operator of order $2m,$ $m\in\mbox{\smallBbb R},$ 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 $\ell_2$ condition number of B satisfies {\rm cond}({\mbox {\bf {\em B}}})\leq C N^{2|m|/d} \left(\frac{h_{\rm max}}{h_{\rm min}}\right)^{d-2m}, where hmax 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 ${\cal O}(N^{2|m|/d})$. 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.