High-order Nystr\"om discretizations for the solution of integral equation formulations of two-dimensional Helmholtz transmission problems

TL;DR

This paper develops and analyzes high-order Nyström discretization methods for solving well-conditioned boundary integral equations in 2D Helmholtz transmission problems, demonstrating stability and convergence including superalgebraic rates for analytic boundaries.

Contribution

It introduces a comprehensive stability and convergence analysis of Nyström methods for multiple integral equation formulations of Helmholtz transmission problems, including new generalized equations.

Findings

Proves stability of Nyström discretizations for classical and new integral equations.

Establishes convergence in Sobolev spaces, ensuring pointwise accuracy.

Shows superalgebraic convergence for analytic boundary cases.

Abstract

We present and analyze fully discrete Nystr\"om methods for the solution of three classes of well conditioned boundary integral equations for the solution of two dimensional scattering problems by homogeneous dielectric scatterers. Specifically, we perform the stability analysis of Nystr\"om discretizations of (1) the classical second kind integral equations for transmission problems [KressRoach, 1978], (2) the single integral equation formulations [Kleinman & Martin,1988], and (3) recently introduced Generalized Combined Source Integral Equations [Boubendir,Bruno, Levadoux and. Turc,2013]. The Nystr\"om method that we use for the discretization of the various integral equations under consideration are based on global trigonometric approximations, splitting of the kernels of integral operators into singular and smooth components, and explicit quadratures of products of singular parts (logarithms) and trigonometric polynomials. The discretization of the integral equations (2) and (3) above requires special care as these formulations feature compositions of boundary integral operators that are pseudodifferential operators of positive and negative orders respectively. We deal with these compositions through Calderon's calculus and we establish the convergence of fully discrete Nystr\"om methods in appropriate Sobolev spaces which implies pointwise convergence of the discrete solutions. In the case of analytic boundaries, we establish superalgebraic convergence of the method.