A Nystrom method for the two dimensional Helmholtz hypersingular equation
Provides a simple, convergent numerical method for solving the Helmholtz hypersingular equation, which is relevant for wave scattering problems in computational acoustics and electromagnetics.
The paper proposes a Nyström discretization for the Helmholtz hypersingular integral equation on smooth planar domains, achieving second-order convergence for two specific parameter choices and first-order otherwise, with numerical validation.
In this paper we propose and analyze a class of simple Nyström 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.