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 (0.1π, 1.9π), 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 Nyström method, which has a number of instability angles in the interval mentioned.

Spline Galerkin Methods for the Double Layer Potential Equations on Contours with Corners

“…Thus, it was discovered that at least for the splines of degree 0, 1 and 2 the corresponding Galerkin method is always stable provided that all opening angles of the corner points are located in the interval [0.1π, 1.9π]. Similar results concerning the spline Galerkin methods for the Sherman-Lauricella equation have been recently obtained in [12]. Note that this effect is in strong contrast with the behaviour of the Nyström method which possesses instability angles in the interval [0.1π, 1.9π], [13].…”

Critical Angles of the Nyström Method for Double Layer Potential Equation

“…[4,5,20]. Its stability for the Sherman-Lauricella and Muskhelishvili equations has been studied earlier [9][10][11]15]. For the double layer potential equation, the stability is proven for sufficiently nice curves -polygons or curves coinciding with polygons in neighbourhoods of corners, or/and for sufficiently nice double layer potential operatorsperturbations of the identity by compact and small norm operators [18,21,22].…”

“…Such angles are called critical and if Γ has a critical angle, the Nyström method is not stable regardless of the shape of the curve. In contrast, spline Galerkin methods do not have any critical angles [15,17] but the complexity of the implementation and computational cost are significantly higher than for Nyström methods. Therefore, in practical computations Nyström type methods are preferable and there are modifications designed to handle the instability induced by non-smooth boundaries [7,19].…”