Rapid solution of first kind boundary integral equations in [formula omitted

Abstract

Weakly singular boundary integral equations (BIEs) of the first kind on polyhedral surfaces Γ in R 3 are discretized by Galerkin BEM on shape-regular, but otherwise unstructured meshes of meshwidth h. Strong ellipticity of the integral operator is shown to give nonsingular stiffness matrices and, for piecewise constant approximations, up to O( h 3) convergence of the farfield. The condition number of the stiffness matrix behaves like O( h −1) in the standard basis. An O( N) agglomeration algorithm for the construction of a multilevel wavelet basis on Γ is introduced resulting in a preconditioner which reduces the condition number to O(| log h|). A class of kernel-independent clustering algorithms (containing the fast multipole method as special case) is introduced for approximate matrix–vector multiplication in O(N( log N) 3) memory and operations. Iterative approximate solution of the linear system by CG or GMRES with wavelet preconditioning and clustering-acceleration of matrix–vector multiplication is shown to yield an approximate solution in log-linear complexity which preserves the O( h 3) convergence of the potentials. Numerical experiments are given which confirm the theory.