A fast multi-level boundary element method for the Helmholtz equation

Most recently, we have developed a novel multi-level boundary element method (MLBEM) for the solution of the steady heat diffusion equation involving asymptotically decaying non-oscillatory log-singular and strongly singular kernels. This hierarchical approach generalizes the pioneering work of Brandt and Lubrecht on multi-level multi-integration (MLMI) and C-cycle multi-grid to the broader class of mixed boundary value problems. The result is a fast, accurate and efficient boundary element method. The present paper extends this new computational methodology to the solution of the Helmholtz equation involving oscillatory log-singular and strongly singular kernels for two-dimensional problems. We consider a direct boundary element formulation and, due to the nature of the fundamental solutions, split the corresponding boundary integral equation into real and imaginary parts. Then, we introduce double-noded corners to facilitate a patch-by-patch application of the MLMI algorithm for fast matrix-vector and matrix-transpose-vector multiplications within bi-conjugate gradient methods. The performance of the proposed fast MLBEM is investigated using a numerical example that possesses an exact solution. For wave numbers κ=20 and below, we demonstrate that the fast MLBEM algorithm for the Helmholtz equation is robust, accurate, and exceptionally efficient.