Spline Galerkin methods for the double layer potential equations on contours with corners

TL;DR

This paper investigates the stability of spline Galerkin methods for double layer potential equations on contours with corners, revealing stability conditions related to corner angles and demonstrating good convergence even with discontinuous data.

Contribution

It introduces a numerical approach to analyze the stability of spline Galerkin methods based on corner angles, contrasting with Nyström methods and providing new stability insights.

Findings

No instability angles in [0.1π, 1.9π] for spline orders 0, 1, 2

Spline Galerkin methods show good convergence despite discontinuities

Stability depends only on corner opening angles, not on contour shape

Abstract

Spline Galerkin methods for the double layer potential equation on contours with corners are studied. The stability of the method depends on the invertibility of some operators $R_{\tau}$ associated with the corner points $\tau$. The operators $R_{\tau}$ do not depend on the shape of the contour but only on the opening angles of the corner points $\tau$. The invertibility of these operators is studied numerically via the stability of the method on model curves, all corner points of which have the same opening angle. The case of the splines of order $0,1$ and $2$ is considered. It is shown that no opening angle located in the interval $[0.1\pi,1.9\pi]$ can cause the instability of the method. This result is in strong contrast with the Nystr{\"o}m method, which has four instability angles in the interval mentioned. Numerical experiments show a good convergence of the methods even if the right-hand side of the equation has discontinuities located at the corner points of the contour.