A multilevel boundary-element method for two-dimensional steady heat diffusion

A fast, accurate, and efficient multilevel boundary-element method (MLBEM) is developed to solve general boundary-value problems arising in computational mechanics. 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 multilevel multi-integration (MLMI) in several important ways to address problems with mixed boundary conditions. We utilize bi-conjugate gradient methods (BCGMs) and implement the MLMI approach for fast matrix and matrix transpose multiplication for every iteration loop. After 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 a broad range of steady-state heat diffusion problems. Here, for a model problem in an L-shaped domain, we demonstrate that the computational complexity of the proposed method approaches the desired goal of N In N, where N is the number of degrees of freedom. For this problem, we show that the MLBEM algorithm reduces computer run times by a factor of 6,295, whereas the memory requirements are reduced 432 times compared to conventional boundary-element methods, while preserving the accuracy of the numerical solution. Furthermore, the method can be extended in a straightforward manner to the solution of many problems in science and engineering that result in very large sets of matrix equations when the associated integral equations are discretized.