A Fast Multi-Level Boundary Element Method for the Steady Heat Diffusion Equation

Abstract

A fast, accurate and efficient multi-level boundary element method is developed to solve general boundary value problems. Here we concentrate on problems of two-dimensional steady potential flow and present a fast direct boundary element formulation. This novel method extends the pioneering work of Brandt and Lubrecht on multi-level multi-integration (MLMI) in several important ways to address problems with mixed boundary conditions. We utilize bi-conjugate gradient methods (BCGM) and implement the MLMI approach for fast matrix and matrix transpose multiplication for every iteration loop. Furthermore, by introducing a C-cycle multigrid algorithm, we find that the number of iterations for the bi-conjugate gradient methods is independent of the boundary element mesh discretization for problems of steady-state heat diffusion considered in this paper. As a result, the computational complexity of the proposed method is proportional to only N · log(N), where N is the number of degrees of freedom.