Fast direct solution of 3-D scattering problems via nonuniform grid-based matrix compression

A fast non-iterative algorithm for the solution of large 3-D acoustic scattering problems is presented. The proposed approach can be used in conjunction with the conventional boundary element discretization of the integral equations of acoustic scattering. The algorithm involves domain decomposition and uses the nonuniform grid (NG) approach for the initial compression of the interactions between each subdomain and the rest of the scatterer. These interactions, represented by the off-diagonal blocks of the boundary element method matrix, are then further compressed while constructing sets of interacting and local basis and testing functions. The compressed matrix is obtained by eliminating the local degrees of freedom through the Schur's complement-based technique procedure applied to the diagonal blocks. In the solution process, the interacting unknowns are first determined by solving the compressed system equations. Subsequently, the local degrees of freedom are determined for each subdomain. The proposed technique effectively reduces the oversampling typically needed when using low-order discretization techniques and provides significant computational savings.