Spline Galerkin methods for the Sherman-Lauricella equation on contours with corners

TL;DR

This paper analyzes spline Galerkin methods for solving the Sherman-Lauricella integral equation on contours with corners, establishing stability conditions, and demonstrating good convergence through numerical experiments.

Contribution

It provides necessary and sufficient conditions for stability of spline Galerkin methods on contours with corners, including new stability results contrasting with Nyström methods.

Findings

Spline Galerkin methods are stable if associated corner operators are invertible.

Numerical experiments confirm good convergence and validate theoretical stability conditions.

Galerkins with splines of order 0, 1, 2 are stable for corners with angles in (0.1π, 1.9π).

Abstract

Spline Galerkin approximation methods for the Sherman-Lauricella integral equation on simple closed piecewise smooth contours are studied, and necessary and sufficient conditions for their stability are obtained. It is shown that the method under consideration is stable if and only if certain operators associated with the corner points of the contour are invertible. Numerical experiments demonstrate a good convergence of the spline Galerkin methods and validate theoretical results. Moreover, it is shown that if all corners of the contour have opening angles located in interval $(0.1\pi, 1.9\pi)$, then the corresponding Galerkin method based on splines of order $0$, $1$ and $2$ is always stable. These results are in strong contrast with the behaviour of the Nystr\"om method, which has a number of instability angles in the interval mentioned.