Proxy-surface-based fast direct solver for TE-mode scattering problems on distributed memory systems
For computational electromagnetics researchers, this work improves accuracy and parallel efficiency of direct solvers for Helmholtz transmission problems, though it is an incremental improvement over existing HSS-based methods.
This paper presents a fast direct solver for TE-mode electromagnetic scattering that achieves O(h^3) convergence, compared to O(h) for conventional HSS solvers, and demonstrates nearly ideal strong and weak scalability 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 Nyström 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.