The stability of the Nystr\"om method for double layer potential equations

TL;DR

This paper investigates the stability conditions of the Nyström method applied to double layer potential equations on smooth contours, identifying key operator invertibility criteria and demonstrating angles causing instability through numerical experiments.

Contribution

It establishes necessary and sufficient conditions for the Nyström method's stability based on operator invertibility related to corner angles.

Findings

Stability depends on invertibility of operators linked to corner angles

Numerical experiments identify specific opening angles causing instability

Provides criteria for ensuring stable application of the Nyström method

Abstract

The stability of the Nystr\"om method for the double layer potential equation on simple closed piecewise smooth contours is studied. Necessary and sufficient conditions of the stability of the method are established. It is shown that the method under consideration is stable if and only if certain operators associated with the opening angles of the corner points are invertible. Numerical experiments show that there are opening angles which cause instability of the method.