H-Matrix Accelerated Direct Matrix Solver for Maxwell's Equations using the Chebyshev-based Nystr\"om Boundary Integral Equation Method

TL;DR

This paper introduces an H-matrix accelerated direct solver for Maxwell's equations that leverages a Chebyshev-based boundary integral method, significantly improving efficiency for large, complex electromagnetic scattering problems.

Contribution

It develops a novel H-matrix framework combined with the Chebyshev-based boundary integral equation method to efficiently solve large-scale Maxwell's equations with complex geometries.

Findings

The solver performs efficiently on electrically large and high contrast dielectric problems.

It outperforms traditional methods like LCN in matrix fill and solution times for large problems.

The approach is robust for complex geometries and near-resonance conditions.

Abstract

An H-matrix accelerated direct solver employing the high-order Chebyshev-based Boundary Integral Equation (CBIE) method has been formulated, tested, and profiled for performance on high contrast dielectric materials and electrically large perfect electric conductor objects. The matrix fill performance of the CBIE proves to be fast for small to moderately sized problems compared to its counterparts, e.g. the locally corrected Nystr\"om (LCN) method, due to the way it handles the singularities by means of a global change of variable method. However, in the case of electrically large scattering problems, the matrix fill and factorization still dominate the solution time when using a direct solution approach. To address this issue, an H-Matrix framework is employed, effectively resolving the challenge and establishing the CBIE as a competitive high-order method for solving scattering problems with poorly conditioned matrix equations. The efficacy of this approach is demonstrated through extensive numerical results, showcasing its robustness to problems that are electrically large, near physical resonances, or that have large dielectric permittivities. The capability of the proposed solver for handling arbitrary geometries is also demonstrated by considering various scattering examples from complex CAD models.