Convergence analysis of a high-order Nystrom integral-equation method for surface scattering problems

TL;DR

This paper provides a convergence analysis for a high-order Nystrom integral-equation method used in 3D surface scattering problems, demonstrating super-algebraic convergence and stability of the scheme.

Contribution

It offers the first theoretical convergence and stability analysis for this high-order Nystrom method applied to 3D acoustic scattering.

Findings

Establishes stability of the method in the L^2 norm.

Derives convergence estimates in L^2 and L^∞ norms.

Confirms super-algebraic convergence for smooth right-hand sides.

Abstract

In this paper we present a convergence analysis for the Nystrom method proposed in [Jour. Comput. Phys. 169 pp. 2921-2934, 2001] for the solution of the combined boundary integral equation formulations of sound-soft acoustic scattering problems in three-dimensional space. This fast and efficient scheme combines FFT techniques and a polar change of variables that cancels out the kernel singularity. We establish the stability of the algorithms in the $L^2$ norm and we derive convergence estimates in both the $L^2$ and $L^\infty$ norms. In particular, our analysis establishes theoretically the previously observed super-algebraic convergence of the method in cases in which the right-hand side is smooth.