On the Stability of the Nyström Method for the Muskhelishvili Equation on Contours with Corners

Abstract

Abstract. The stability of the Nyström method for the Muskhelishvili equation on piecewise smooth simple contours Γ is studied. It is shown that in the space L 2 the method is stable if and only if certain operators Aτ j from an algebra of Toeplitz operators are invertible. The operators Aτ j depend on the parameters of the equation considered, on the opening angles θ j of the corner points τ j ∈ Γ and on parameters of the approximation method mentioned. Numerical experiments show that there are opening angles where the operators Aτ j are non-invertible. Therefore, for contours with such corners the method under consideration is not stable. Otherwise, the method is always stable. Numerical examples show an excellent convergence of the method.

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

“…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].…”

“…Nevertheless, the approximate solutions presented in Figure 2 demonstrate a good accuracy. We also computed Galerkin method solutions of the double layer potential equation with the right-hand sides and curves from [13]. Although these results are not reported here, there is a good correlation with approximate solutions of [13] obtained by the Nyström method.…”

“…However, these operators belong to an algebra of Toeplitz operators generated by piecewise continuous matrix functions and at present there is no analytic tool to check their invertibility. On the other hand, a numerical approach to such a kind of problem has been proposed in [10,11]. Thus one can consider stability of an approximation method on curves having corner points with the same opening angle.…”

“…As a result, at any critical point of the method, the graph representing the condition numbers has to have an "infinite" peak regardless of the family of the curves used. In this paper we employ the curves L 1 (θ), L 2 (θ), proposed in [10,11], which have one and two corner points, respectively, together with a new 4-corner curve L 4 (θ). The curves L 1 (θ), L 2 (θ) have the following parametrizations where O denotes the origin, − → t A and − → t B are, respectively, the tangential vectors at the points A and B, and ( − → t 1 , − → t 2 ) is the angle measured from − → t 1 to − → t 2 in the counterclockwise direction (see Figure 3).…”

“…However, for contours with corners, the stability analysis of the methods is not complete. On the other hand, it is known that for boundary integral equations the presence of corners on the boundary may lead to extra conditions required for the stability of the approximation method considered [9,10,11,13]. The aim of the present work is twofold.…”