Taiwanese Journal of Mathematics


Jin Huang, Zi Cai Li, Tao L¨u, and Rui Zhu

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To solve the boundary integral equations (BIE) of mixed boundary conditions, we propose the mechanical quadrature methods (MQM) using specific quadrature rule to deal with weakly singular integrals. Denote $h_{m}$ as the mesh width of a curved edge $\Gamma_{m}$ ($m=1,...,d)$ of polygons. Then the multivariate asymptotic expansions of solution errors are found to be $O(h^{3}),$ where $h=\max_{1\leq m\leq d}h_{m}.$ Hence, by using the splitting extrapolation methods (SEM), the high convergence rates as $O(h^{5})$ can be achieved. Moreover, numerical examples are provided to support our theoretical analysis.

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Taiwanese J. Math., Volume 12, Number 9 (2008), 2341-2361.

First available in Project Euclid: 18 July 2017

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Primary: 65R20: Integral equations 45L10

mixed boundary condition polygon mechanical quadrature method splitting extrapolation boundary integral equation


Huang, Jin; Li, Zi Cai; L¨u, Tao; Zhu, Rui. SPLITTING EXTRAPOLATIONS FOR SOLVING BOUNDARY INTEGRAL EQUATIONS OF MIXED BOUNDARY CONDITIONS ON POLYGONS BY MECHANICAL QUADRATURE METHODS. Taiwanese J. Math. 12 (2008), no. 9, 2341--2361. doi:10.11650/twjm/1500405183. https://projecteuclid.org/euclid.twjm/1500405183

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  • K. Atkinson, The numerical solution of Laplace's equation on a wedge, IMA. J Num. Anal., 4 (1984), 19-41.
  • K. E. Atkinson and G. Chandler, Boundary integral equation methods for solving Laplace's equation with nonlinear boundary conditions: The smooth boundary case, Math. Comp., 55 (1990), 451-472.
  • P. M. Anselone, Collectively Compact Operator Approximation Theory, Prentice-Hall, Englewood Cliffs, New Jersey, 1971.
  • C. A. Brebbia, Boundary Element Methods, Butterworths, 1980.
  • G. Chandler, Galerkin's method for boundary integral equations on polygonal domains, Aust. Math Soc. Ser., 26 (1984), 1-13.
  • F. Chatelin, Spectral Approximation of Linear Operator, Academic Press, 1983.
  • Z. Daniel, Standard Mathematical Tables and Formulae, New York, 1996.
  • P. Davis, Methods of Numerical Integration, Second, Academic Press, New York, 1984.
  • D. L. Dwoyer, M. Y. Hussaini and R. G. Voigt, Finite Elements Theory and Application, Springer-Verlag, New York, 1986.
  • A. S. Householder, The Theory of Matrices in Numerical Analysis, Blaisdell Publishing Company, 1964.
  • J. Huang and T. Lü, The mechanical quadrature methods and their splitting extrapolations for solving first-kind boundary integral equations on polygonal regions (in Chinese), Math. Num. Sinica, 1 (2004), 51-60.
  • R. Kress, Linear Integral Equations, Springer-Verlag, 1989.
  • C. B. Lin, T. Lü and T. M. Shih, The Splitting Extrapolation Method, World Scientific, Singapore, 1995.
  • Q. Lin and T. Lü, Splitting extrapolation for multidimensional problem, J. Comp. Math., 1 (1983), 376-383.
  • T. Lü and J. Huang, Quadrature methods with high accuracy and extrapolation for solving boundary integral equations of first-kind (in Chinese), Math Num. Sinica, 1 (2000), 59-72.
  • T. Lü and J. Lu, Splitting extrapolation for solving the second order elliptic system with curved boundary in R$^{d}$ by using d-quadratic isoparametric finite element, Appl. Numer. Math., 40 (2002), 467-481.
  • F. Paris and J. Cans, Boundary Element Method, Oxford University Press, 1997.
  • U. Rüde and A. Zhou, Multi-parameter extrapolation methods for boundary integral equations, Advances in Computational Mathematics, 9 (1998), 173-190.
  • K. Ruotsalainen and W. W Endland, On the boundary element method for some nonlinear boundary value problems, Numer. Math., 53 (1988), 299-314.
  • A. Sidi and M. Israrli, Quadrature methods for periodic singular Fredholm integral equation, J. of Sci Comp., 3 (1988), 201-231.
  • A. Sidi, A new variable transformation for numerical integration, I. S., Num Math., 112 (1993), 359-373.
  • I. H. Sloan and A. Spence, The Galerkin method for integral equations of first-kind with logarithmic kernel, IMA J. Numer. Anal., V (1988), 105-122.
  • J. H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University Press, 1965.
  • Y. S. Xu and Y. H. Zhao, An extrapolation method for a class of boundary integral equations, Math. Comp., 65 (1996), 587-610.
  • Y. Yi, The collocation method for first-kind boundary integral equations on polygonal regions, Math Comp., 189 (1990), 139-154.