High order Nystr\"om methods for transmission problems for Helmholtz equation

TL;DR

This paper introduces superalgebraic compatible Nyström discretizations for Helmholtz boundary operators, ensuring stable and superalgebraic convergence for integral equation solutions on smooth closed curves in 2D.

Contribution

The paper develops a new Fourier-based Nyström discretization method for Helmholtz boundary operators that guarantees superalgebraic convergence and stability.

Findings

Discrete operators converge to continuous ones in Sobolev norms

Nyström discretizations are stable and convergent for Helmholtz equations

Convergence is superalgebraic for smooth solutions

Abstract

We present superalgebraic compatible Nystr\"om discretizations for the four Helmholtz boundary operators of Calder\'{o}n's calculus on smooth closed curves in 2D. These discretizations are based on appropriate splitting of the kernels combined with very accurate product-quadrature rules for the different singularities that such kernels present. A Fourier based analysis shows that the four discrete operators converge to the continuous ones in appropriate Sobolev norms. This proves that Nystr\"om discretizations of many popular integral equation formulations for Helmholtz equations are stable and convergent. The convergence is actually superalgebraic for smooth solutions.