Extrapolation algorithms for solving nonlinear boundary integral equations by mechanical quadrature methods

We study the numerical solution procedure for two-dimensional Laplace’s equation subjecting to non-linear boundary conditions. Based on the potential theory, the problem can be converted into a nonlinear boundary integral equations. Mechanical quadrature methods are presented for solving the equations, which possess high accuracy order O(h3) and low computing complexities. Moreover, the algorithms of the mechanical quadrature methods are simple without any integration computation. Harnessing the asymptotical compact theory and Stepleman theorem, an asymptotic expansion of the errors with odd powers is shown. Based on the asymptotic expansion, the h3 −Richardson extrapolation algorithms are used and the accuracy order is improved to O(h5). The efficiency of the algorithms is illustrated by numerical examples.

[1]  Jin Huang,et al.  High accuracy eigensolution and its extrapolation for potential equations , 2010 .

[2]  Mark A. Kelmanson Solution of Nonlinear Elliptic Equations with Boundary Singularities by an Integral Equation Method , 1984 .

[3]  Moshe Israeli,et al.  Quadrature methods for periodic singular and weakly singular Fredholm integral equations , 1988, J. Sci. Comput..

[4]  Jukka Saranen,et al.  On the collocation method for a nonlinear boundary integral equation , 1989 .

[5]  Zhu Wang,et al.  Extrapolation Algorithms for Solving Mixed Boundary Integral Equations of the Helmholtz Equation by Mechanical Quadrature Methods , 2009, SIAM J. Sci. Comput..

[6]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[7]  G. Chandler,et al.  Galerkin's method for boundary integral equations on polygonal domains , 1984, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[8]  JinHuang,et al.  THE MECHANICAL QUADRATURE METHODS AND THEIR EXTRAPOLATION FOR SOLVING BIE OF STEKLOV EIGENVALUE PROBLEMS , 2004 .

[9]  Yuesheng Xu,et al.  An extrapolation method for a class of boundary integral equations , 1996, Math. Comput..

[10]  Eckart Schnack,et al.  A hybrid coupled finite-boundary element method in elasticity , 1999 .

[11]  Khosrow Maleknejad,et al.  Numerical method for a nonlinear boundary integral equation , 2006, Appl. Math. Comput..

[12]  P. K. Banerjee The Boundary Element Methods in Engineering , 1994 .

[13]  Rainer Kress,et al.  A Nyström method for boundary integral equations in domains with corners , 1990 .

[14]  Boundary element technique : C.A. Brebbia, J.C.F. Tellas and L.C. Wrobel Springer-Verlag, Berlin, 1984, 464 pp., US $70.20; DM 188 , 1984 .

[15]  K. Ruotsalainen,et al.  On the boundary element method for some nonlinear boundary value problems , 1988 .

[16]  Kendall E. Atkinson,et al.  BOUNDARY INTEGRAL EQUATION METHODS FOR SOLVING LAPLACE'S EQUATION WITH NONLINEAR BOUNDARY CONDITIONS: THE SMOOTH BOUNDARY CASE , 1990 .

[17]  Tao Lü,et al.  Mechanical quadrature methods and their splitting extrapolations for boundary integral equations of first kind on open arcs , 2009 .

[18]  Andrzej J. Nowak,et al.  Boundary value problems in heat conduction with nonlinear material and nonlinear boundary conditions , 1981 .

[19]  C. Brebbia,et al.  The Boundary Element Technique for the Analysis of Automotive Structures , 1984 .

[20]  C. Brebbia,et al.  Boundary Element Techniques , 1984 .