Low-rank properties in the hybrid finite element-boundary integral method

The iterative solution of the hybrid finite element-boundary integral equation (FE-BIE) system can be accelerated by using a multilevel fast multipole algorithm (MLFMA) to accelerate the BIE interactions. The far interactions in the MLFMA are of low complexity, which can be traced back to the original dense BIE matrix where off-diagonal submatrices are of low-rank, provided the unknowns are properly ordered. To accelerate the FE part, we investigate the existence of similar low-rank properties in the FE matrix. Previously, they were shown to exist in the FE discretization matrices of elliptic partial differential equations, but not for the non-elliptic time harmonic system of Maxwell´s equations. Comparing the Schur complement form of the FE system with a discrete Poincaré-Steklov operator, the existence of such low-rank properties is suggested, and this theory is confirmed experimentally by means of a numerical example.