TY - JOUR
T1 - Spline Galerkin Methods for the Sherman-Lauricella Equation on Contours with Corners
AU - Didenko, Victor D.
AU - Tang, Tao
AU - Vu, Anh My
N1 - Funding information:
Faculty of Science, Universiti Brunei Darussalam, Bandar Seri Begawan, BE1410, Brunei ([email protected], [email protected]). The research of these authors was partially supported by the Universiti Brunei Darussalam under grant UBD/GSR/S&T/19.
*Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong ([email protected]). Current address: Department of Mathematics, South University of Science and Technology, Shenzhen, Guangdong 518055, China ([email protected]). The research of this author was partially supported by Hong Kong Research Grants Council CERG grants, the National Science Foundation of China, and Hong Kong Baptist University FRG grants.
Publisher copyright:
© 2015, Society for Industrial and Applied Mathematics
PY - 2015/12/17
Y1 - 2015/12/17
N2 - 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, or 2 is always stable. These results are in strong contrast with the behavior of the Nystrom method, which has a number of instability angles in the interval mentioned.
AB - 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, or 2 is always stable. These results are in strong contrast with the behavior of the Nystrom method, which has a number of instability angles in the interval mentioned.
KW - Critical angles
KW - Sherman-Lauricella equation
KW - Spline Galerkin method
KW - Stability
UR - http://www.scopus.com/inward/record.url?scp=84973293527&partnerID=8YFLogxK
U2 - 10.1137/140997968
DO - 10.1137/140997968
M3 - Journal article
AN - SCOPUS:84973293527
SN - 0036-1429
VL - 53
SP - 2752
EP - 2770
JO - SIAM Journal on Numerical Analysis
JF - SIAM Journal on Numerical Analysis
IS - 6
ER -