Fast direct solution of 3-D volume integral equations by skeletonization for dynamic electromagnetic wave problems

The skeletonization-based direct solver is extended to the method of moments (MoM) solution of 3-D volume integral equations (VIEs) for dynamic electromagnetic wave problems. The direct solver approximates the impedance matrix by the hierarchical data sparse representation through skeletonization and then obtains the inverse matrix implicitly. Improvement is made in the proxy strategy to accelerate the skeletonization. Detailed studies are presented on the behavior of the direct solver with respect to target shape, mesh density, and permittivity. It is revealed that the fast direct solver is promising for nano applications, especially for the cases where large and high-contrast objects are involved, because iterative solvers always fail in reaching convergence due to the associated ill-conditioned impedance matrix.