Application of the adaptive cross approximation technique for the coupled BE-FE solution of symmetric electromagnetic problems

Abstract

Electromagnetic devices can be analysed by the coupled BE-FE method, where the conducting and magnetic parts are discretized by finite elements. In contrast, the surrounding space is described with the help of the boundary element method (BEM). This discretization scheme is well suited especially for problems including moving parts (see [12]). The BEM discretization of the boundary integral operators usually leads to dense matrices without any structure. A naive strategy for the solution of the corresponding linear system would need at least O(N ²) operations and memory, where N ist the number of unknowns. Methods such as fast multipole [6] and panel clustering [9] provide an approximation to the matrix in almost linear complexity. These methods are based on explicitly given kernel approximations by degenerate kernels, i.e. a finite sum of separable functions, which may be seen as a blockwise low-rank approximation of the system matrix. The blockwise approximant permits a fast matrix-vector multiplication, which can be exploited in iterative solvers, and can be stored efficiently. In contrast to the methods mentioned we will generate [2] the low-rank approximant from the matrix itself using only few entries and without using any explicit a priori known degenerate-kernel approximation. Special emphasis is put on the handling of symmetry conditions in connection with ACA. The feasibility of the proposed method is demonstrated by means of a numerical example.