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

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 where the density function—that is, the solution to the integral equation—exhibits unbounded singular behavior at edges and corners. While such difficulties can be circumvented in two-dimensional configurations, they constitute an intrinsic feature of existing three-dimensional Maxwell integral formulations, in which the tangential component of the surface current density diverges along edges. 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, can be generalized 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.