A Numerical Study on the Compressibility of Subblocks of Schur Complement Matrices Obtained from Discretized Helmholtz Equations

The compressibility of Schur complement matrices is the essential ingredient for H-matrix techniques, and is well understood for Laplace type problems. The Helmholtz case is more difficult: there are several theoretical results which indicate when good compression is possible with additional techniques, and in practice sometimes basic H-matrix techniques work well. We investigate the compressibility here with extensive numerical experiments based on the SVD. We find that with growing wave number k, the ǫ-rank of blocks corresponding to a fixed size in physical space of the Green’s function is always growing like O(k), with α ∈ [ 3 4 , 1] in 2d and α ∈ [ 4 3 , 2] in 3d.