A Chebyshev-based High-order-accurate Integral Equation Solver for Maxwell's Equations

TL;DR

This paper presents a high-order accurate integral equation solver for Maxwell's equations using Chebyshev polynomials, achieving spectral accuracy on smooth surfaces and demonstrating superior convergence compared to traditional methods.

Contribution

A novel hybrid Nyström-collocation method employing Chebyshev polynomials for discretizing Maxwell's equations with spectral accuracy on complex geometries.

Findings

Spectral convergence demonstrated on various geometries.

Outperforms traditional RWG-based Method-of-Moments solver.

Effective for both metallic and dielectric scattering problems.

Abstract

This paper introduces a new method for discretizing and solving integral equation formulations of Maxwell's equations which achieves spectral accuracy for smooth surfaces. The approach is based on a hybrid Nystr\"om-collocation method using Chebyshev polynomials to expand the unknown current densities over curvilinear quadrilateral surface patches. As an example, the proposed strategy is applied the to Magnetic Field Integral Equation (MFIE) and the N-M\"uller formulation for scattering from metallic and dielectric objects, respectively. The convergence is studied for several different geometries, including spheres, cubes, and complex NURBS geometries imported from CAD software, and the results are compared against a commercial Method-of-Moments solver using RWG basis functions.