Stability of the Nyström Method for the Sherman–Lauricella Equation

East Asian J. Appl. Math.

Helsing

2011

Self Cite

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.

“…(2.5) have been obtained in Ref. [3]. For the convenience of the reader, we reformulate the corresponding result as follows.…”

Section: The Nyström Methods and The Operators A C Jmentioning

confidence: 93%

“…According to [3], the approximate values ω(τ l p ) of an exact solution ω of Eq. (1.1) at the points τ l p are defined by the following system of algebraic equations:…”

Section: The Nyström Methods and The Operators A C Jmentioning

confidence: 99%

“…The operator (2.4) is called the zero average displacement choice of T S L [8], and the role of correcting operators T S L is discussed in [3,7]. …”

Section: The Nyström Methods and The Operators A C Jmentioning

confidence: 99%

“…[3]. From a purely practical view- point, however, there may be even better approximation strategies to solve the ShermanLauricella equation (1.1) on piecewise smooth contours.…”

Section: The Operator a C J Is Invertible If And Only If The Sequence (Amentioning

confidence: 99%

“…[3], and find that in the interval (0.1π, 1.9π) there are angles for which the operators A c j are not invertible.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

East Asian J. Appl. Math.

Helsing

2011

Self Cite

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

Proc Appl Math and Mech

Helsing

2013

Self Cite

This paper deals with approximate solutions to integral equations arising in boundary value problems for the biharmonic equation in simply connected piecewise smooth domains. The approximation method considered demonstrates excellent convergence even in the case of boundary conditions discontinuous at corner points. In an application we obtain very accurate approximations for some characteristics of two-dimensional Stokes flow in non-smooth domains.

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

Recent Trends in Operator Theory and Partial Differential Equations

2017

Self Cite

Spline Galerkin methods for the double layer potential equation on contours with corners are studied. The stability of the method depends on the invertibility of some operators R τ associated with the corner points τ . The operators R τ do not depend on the shape of the contour but only on the opening angles of the corner points τ . The invertibility of these operators is studied numerically via the stability of the method on model curves, all corner points of which have the same opening angle. The case of the splines of order 0, 1 and 2 is considered. It is shown that no opening angle located in the interval [0.1π, 1.9π] can cause the instability of the method. This result is in strong contrast with the Nyström method, which has four instability angles in the interval mentioned. Numerical experiments show a good convergence of the methods even if the right-hand side of the equation has discontinuities located at the corner points of the contour.