Features of the Nyström Method for the Sherman-Lauricella Equation on Piecewise Smooth Contours

Abstract

Abstract. The stability of the Nyström method for the Sherman-Lauricella equation on contours with corner points c j , j = 0, 1, . . . , m relies on the invertibility of certain operators A c j belonging to an algebra of Toeplitz operators. The operators A c j do not depend on the shape of the contour, but on the opening angle θ j of the corresponding corner c j and on parameters of the approximation method mentioned. They have a complicated structure and there is no analytic tool to verify their invertibility. To study this problem, the original Nyström method is applied to the Sherman-Lauricella equation on a special model contour that has only one corner point with varying opening angle θ j . In the interval (0.1π, 1.9π), it is found that there are 8 values of θ j where the invertibility of the operator A c j may fail, so the corresponding original Nyström method on any contour with corner points of such magnitude cannot be stable and requires modification.

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

“…In other words, Figure 4.1 supports the fact that the invertibility of the associated operators A τ depends on the opening angle but not on the shape of the contour. It is remarkable that the above instability angles coincide up to three digits with the corresponding instability angles of the Nyström method for the Sherman-Lauricella equation [3]. It leads to a conjecture that if k = 1, then the Nyström method for the Muskhelishvili equation is stable if and only if so is for the Sherman-Lauricella equation.…”

Approximate solution of boundary integral equations for biharmonic problems in non‐smooth domains

“…Consider the operators B θ j ,δ,ε defined on the space l 2 by [6, Formula (3.9)]. It was noted in [7] that there is at most countable set S c of opening angles θ j ∈ (0, 2π), called critical angles, such that the operator B θ j ,δ,ε is not invertible.…”

“…Moreover, since f ∈ W 1 2 (Γ, c 0 , c 1 , c 2 , c 3 ) and condition (5) is satisfied, the approximate solutions ω n converge to an exact solution of the corresponding equation (2). Besides, all opening angles θ j of the corner points are equal to π/2, and by [7] the angle π/2 / ∈ S c . Therefore, the above method is stable and converges for any right hand side from the above-mentioned space.…”

“…Let us compute the approximate solution using the above algorithm. By [5, Lemma 5.1.3] the problem is reduced to (7) with the right-hand side f (t) = 4x 3 − 12xy 2 + i(4y 3 − 12x 2 y). Once an approximate solution ω n of (2) is obtained from (4), an approximation (ũ i ,ṽ i ) to the velocity field (u i , v i ) is computed via (10) at 90, 000 points τ i on a Cartesian grid on D.…”

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

“…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).…”