A Nystrom method for the two dimensional Helmholtz hypersingular equation

TL;DR

This paper introduces a simple Nyström discretization method for the 2D Helmholtz hypersingular equation, demonstrating convergence of order two for specific parameter choices and providing numerical validation.

Contribution

The paper proposes a novel class of Nyström discretizations for the Helmholtz hypersingular equation, analyzing their convergence properties and identifying parameter choices that yield higher-order accuracy.

Findings

Two parameter choices produce second-order convergent methods.

All other stable methods achieve first-order convergence.

Numerical experiments confirm the theoretical convergence rates.

Abstract

In this paper we propose and analyze a class of simple Nystr\"om discretizations of the hypersingular integral equation for the Helmholtz problem on domains of the plane with smooth parametrizable boundary. The method depends on a parameter (related to the staggering of two underlying grids) and we show that two choices of this parameter produce convergent methods of order two, while all other stable methods provide methods of order one. Convergence is shown for the density (in uniform norm) and for the potential postprocessing of the solution. Some numerical experiments are given to illustrate the performance of the method.