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.
A Fast Multi-Level Boundary Element Method for the Steady Heat Diffusion Equation
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Grigoriev, MM, & Dargush, GF. "A Fast Multi-Level Boundary Element Method for the Steady Heat Diffusion Equation." Proceedings of the ASME 2003 Heat Transfer Summer Conference. Heat Transfer: Volume 3. Las Vegas, Nevada, USA. July 21–23, 2003. pp. 865-873. ASME. https://doi.org/10.1115/HT2003-47450
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