EXPERIENCES WITH THE CONVOLUTION QUADRATURE METHOD AND THE ADAPTIVE CROSS APPROXIMATION IN BEM

Abstract. Wave propagation phenomena occur in reality often in semi-infinite regions. It is well known that such problems can be handled well with the Boundary Element Method (BEM). Focusing on wave propagation problems a formulation in time domain is preferable. But, it is also well known that the BEM with its dense matrices becomes prohibitive with respect to storage and computing time. Several approaches have been developed to overcome these drawbacks. Approaches, such as Fast Multipole gain their efficiency basically from an analytic kernel approximation. The present work focuses on a purely algebraic approach, the adaptive cross approximation (ACA) [2], which was developed for elliptic problems. Indeed, time dependent elastic problems are not elliptic. This conflict is resolved by introducing the well known Convolution Quadrature Method [3, 4] as time stepping scheme. Thus the solution of the time dependent problem ends up in the solution of a system of decoupled Laplace domain problems [1]. This detour is worth since the resulting problems are again elliptic and the ACA can therefore be used in its standard fashion. By means of a geometrical clustering and a reliable admissibility condition, first, a so called hierarchical matrix structure is set up. Then each admissible block can be represented by a low-rank approximation. Results are presented for a symmetric Galerkin formulation in elastodynamics. The stabil-ity behavior is not touched by the compression or the iterative equation solver if a sufficient precision is chosen. Further, as expected, in the high frequency range, the compression rate has to be improved. But, the main advantage of this approach of accelerating a time depen-dent BEM is that it can be easily applied to other fundamental solutions as, e.g., visco- or poroelasticity [5]