A multi-level multi-integral algorithm for the Helmholtz equation

Previous work on the development of multi-level boundary element methods (MLBEM) has produced dramatic computational efficiencies compared to traditional methods. However, for problems involving the oscillatory kernels associated with the Helmholtz equation, the performance improvements demonstrated to date, have been limited due to the presence of coarse mesh Nyquist constraints. In the present paper, we examine an extension of the original Helmholtz MLBEM to remove these limitations, based on a polar decomposition of the kernels and field variables originally proposed by Brandt. In this initial numerical implementation of these ideas, the algorithm is applied to a pair of two-dimensional half-space problems and, therefore, is limited to the fast evaluation of Rayleigh integrals. In general, excellent performance is achieved. For example, in the second problem with a non-dimensional wave number equal to 10⁵, we obtain speed-up factors of approximately 300,000 versus a traditional BEM approach, while maintaining comparable levels of accuracy.