A convergence theorem for a class of Nystrom methods for weakly singular integral equations on surfaces in R^3

TL;DR

This paper proves a convergence theorem for a class of Nystrom methods that use local polynomial corrections to solve weakly singular integral equations on surfaces in three dimensions, improving understanding of their accuracy.

Contribution

It introduces a new convergence theorem for Nystrom methods with polynomial corrections, showing their effectiveness and providing error control under minimal regularity assumptions.

Findings

Convergence depends on problem regularity, polynomial degree, and quadrature order.

Polynomial correction effectively removes weak singularity.

Simple singularity subtraction method is also proven to be convergent.

Abstract

A convergence theorem is proved for a class of Nystrom methods for weakly singular integral equations on surfaces in three dimensions. Fredholm equations of the second kind as arise in connection with linear elliptic boundary value problems for scalar and vector fields are considered. In contrast to methods based on product integration, coordinate transformations and singularity subtraction, the family of Nystrom methods studied here is based on a local polynomial correction determined by an auxiliary system of moment equations. The polynomial correction is shown to remove the weak singularity in the integral equation and provide control over the approximation error. Convergence results for the family of methods are established under minimal regularity assumptions consistent with classic potential theory. Rates of convergence are shown to depend on the regularity of the problem, the degree of the polynomial correction, and the order of the quadrature rule employed in the discretization. As a corollary, a simple method based on singularity subtraction which has been employed by many authors is shown to be convergent.