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

Didenko, Victor D.; Helsing, Johan

doi:10.1002/pamm.201310212

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2015

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“…For piecewise smooth contours, conditions of the stability of the Nyström method are established in [6,7]. These results have been used in [8] in order to construct a very accurate numerical method to find solutions of the biharmonic problem (1.1) in piecewise smooth domains in the case of piecewise continuous boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

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

Didenko¹,

Tang²,

Vu³

2015

SIAM J. Numer. Anal.

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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.

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Section: Introductionmentioning

confidence: 99%