High order-accurate solution of scattering integral equations with unbounded solutions at corners

TL;DR

This paper introduces high-order methods for solving 2D Maxwell integral equations with unbounded solutions at corners, achieving high accuracy without special basis functions or geometry analysis, and with potential extension to 3D problems.

Contribution

It presents novel high-order techniques that handle unbounded densities at corners in integral equations, improving accuracy in challenging geometries without specialized basis functions.

Findings

High-order convergence achieved near corners and edges.

Methods work without prior geometric analysis or singular basis functions.

Potential applicability to 3D Maxwell integral formulations.

Abstract

Although high-order Maxwell integral equation solvers provide significant advantages in terms of speed and accuracy over corresponding low-order integral methods, their performance significantly degrades in presence of non-smooth geometries--owing to field enhancement and singularities that arise at sharp edges and corners which, if left untreated, give rise to significant accuracy losses. The problem is particularly challenging in cases in which the "density" (i.e., the solution of the integral equation) tends to infinity at corners and edges--a difficulty that can be bypassed for 2D configurations, but which is unavoidable in 3D Maxwell integral formulations, wherein the component tangential to an edge of the electrical-current integral density vector tends to infinity at the edge. In order to tackle the problem this paper restricts attention to the simplest context in which the unbounded-density difficulty arises, namely, integral formulations in 2D space whose integral density blows up at corners; the strategies proposed, however, generalize directly to the 3D context. The novel methodologies presented in this paper yield high-order convergence for such challenging equations and achieve highly accurate solutions (even near edges and corners) without requiring a priori analysis of the geometry or use of singular bases.