Spline Galerkin Methods for the Sherman--Lauricella Equation on Contours with Corners

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.