A HIGH ORDER FAST MULTIPOLE BOUNDARY ELEMENT METHOD

The Boundary Element Method can be used to solve the Helmholtz equation in three dimensions. Just the surface has to be discretized, but a solution for the complete domain is obtained. Especially for exterior domains this is an enormous advantage. A drawback of the solution procedure, however is its quadratic complexity. The Fast Multipole Method results in a quasi linear complexity due to an approximation of a matrix vector product and is therefore applicable for solving large scale systems. The variables on the surface are approximated by ansatz functions of a certain order. Typically constant, linear or quadratic element orders are used. In this contribution formulations for arbitrary ansatz functions are presented. Since an application to exterior cases leads to the problem of fictitious frequencies, a Burton Miller formulation is embedded. The presented Collocation Method can treat the hypersingular kernels for arbitrary ansatz functions correctly. The theoretical background is introduced and numerical examples show the performance of the formulation.