Proxy-surface-based fast direct solver for TE-mode scattering problems on distributed memory systems

This paper describes an MPI/OpenMP hybrid parallelized fast direct solver for the scattering problem of transverse electric (TE)-mode electromagnetic waves. Because TE-mode scattering can be reduced to the two-dimensional Helmholtz equation, solvers based on the hierarchically semiseparable (HSS) representation are highly attractive due to their high parallel efficiency. However, as the HSS representation applies low-rank approximations to all off-diagonal blocks, it exhibits poor compatibility with high-order discretization methods. We developed a fast direct solver with $O(h^3)$ convergence for Helmholtz transmission problems, whereas conventional HSS solvers typically yield only $O(h)$ convergence (where $h$ represents intervals between the quadrature nodes). It is based on the weakly singular Burton-Miller boundary integral equation and the Nystr\"om method with a one-point correction. Furthermore, recognizing that matrix component calculation, rather than matrix factorization, dominates the total computational time of HSS-type boundary integral solvers, we introduced a load-balancing method to maximize parallel efficiency. Numerical results demonstrate that the direct solver achieves high-accuracy convergence and nearly ideal strong and weak scalabilities.