Developments of multi-level boundary element methods for steady heat diffusion problems

We have recently developed a novel multi-level boundary element method (MLBEM) for steady heat diffusion in irregular two-dimensional domains. This paper extends the MLBEM methodology to dramatically improve the performance of the original multi-level formulation. First, we perform analyses of numerical error and computational complexity for the multi-level boundary element algorithm and show that the optimal complexity of the algorithm is O(N log N). Next, we consider a model problem of line multi-integral evaluation and investigate the performance of the MLBEM formulation using a single-patch approach. Then we study the performance of the multi-level boundary element formulation on an example Neumann problem of steady heat diffusion leading to a boundary integral equation of the second kind. Here, we solve a problem involving four million degrees of freedom in less than one hour on a desktop workstation. Finally, we consider a model problem in a unit square with mixed boundary conditions and study the performance for the new MLBEM formulation.