3D-ACA for the time domain boundary element method: Comparison of FMM and H-matrix based approaches

The time domain Boundary Element Method (BEM) for the homogeneous wave equation with vanishing initial conditions is considered. The generalized convolution quadrature method (gCQ) developed by Lopez-Fernandez and Sauter is used for the temporal discretisation. The spatial discretisation is done classically using low order shape functions. A collocation approach is applied. Essentially, the gCQ requires to establish boundary element matrices of the corresponding elliptic problem in Laplace domain at several complex frequencies. Consequently, an array of system matrices is obtained. This array of system matrices can be interpreted as a threedimensional array of data which should be approximated by a data-sparse representation. The generalised Adaptive Cross Approximation (3D-ACA) can be applied to get a data sparse representation of these three-dimensional data arrays. Adaptively, the rank of the three-dimensional data array is increased until a prescribed accuracy is obtained. On a pure algebraic level it is decided whether a low-rank approximation of the three-dimensional data array is close enough to the original matrix. Within the data slices corresponding to the BEM calculations at each frequency either the standard H-matrices approach with ACA or a fast multipole (FMM) approach can be used. The third dimension of the data array represents the complex frequencies. Hence, the algorithm makes not only a data sparse approximation in the two spatial dimensions but detects adaptively how much frequencies are necessary for which matrix block. Numerical studies show the performance of these methods.