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Symmetric boundary knot method. (English) Zbl 1006.65500
Summary: The boundary knot method (BKM) is a recent boundary-type radial basis function (RBF) collocation scheme for general partial differential equations. Like the method of fundamental solution (MFS), the RBF is employed to approximate the inhomogeneous terms via the dual reciprocity principle. Unlike the MFS, the method uses a non-singular general solution instead of a singular fundamental solution to evaluate the homogeneous solution so as to circumvent the controversial artificial boundary outside the physical domain. The BKM is meshfree, super-convergent, integration-free, very easy to learn and program. The original BKM, however, loses symmetricity in the presence of mixed boundary. In this study, by analogy with Fasshauer’s Hermite RBF interpolation, we developed a symmetric BKM scheme. The accuracy and efficiency of the symmetric BKM are also numerically validated in some 2D and 3D Helmholtz and diffusion-reaction problems under complicated geometries.

65N38 Boundary element methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
Full Text: DOI
[1] Nardini, D.; Brebbia, C.A., A new approach to free vibration analysis using boundary elements, Appl math modeling, 7, 157-162, (1983) · Zbl 0545.73078
[2] Kansa, E.J., Multiquadrics: a scattered data approximation scheme with applications to computational fluid-dynamics, Comput math appl, 19, 147-161, (1990) · Zbl 0850.76048
[3] Golberg, M.A.; Chen, C.S., The method of fundamental solutions for potential, Helmholtz and diffusion problems, (), 103-176 · Zbl 0945.65130
[4] Kitagawa, T., Asymptotic stability of the fundamental solution method, J comput appl math, 38, 263-269, (1991) · Zbl 0752.65077
[5] Balakrishnan, K.; Ramachandran, P.A., The method of fundamental solutions for linear diffusion-reaction equations, Math comput modeling, 31, 221-237, (2001) · Zbl 1042.35569
[6] Chen, W.; Tanaka, M., A meshless, exponential convergence, integration-free, and boundary-only RBF technique, Comput math appl, (2002), in press · Zbl 0999.65142
[7] Chen, W.; Tanaka, M., New insights into boundary-only and domain-type RBF methods, Int J nonlinear sci numer simulat, 1, 3, 145-151, (2000) · Zbl 0954.65084
[8] Chen W. New RBF collocation schemes and kernel RBF with their applications. International Workshop for Meshfree Methods for Partial Differential Equations. Bonn, Germany; September 2001.
[9] Chen W, Hon YC. Numerical convergence of boundary knot method in the analysis of Helmholtz, modified Helmholtz, and convection-diffusion problems. Submitted for publication. · Zbl 1050.76040
[10] Hon, Y.C.; Chen, W., Boundary knot method for 2D and 3D Helmholtz and convection – diffusion problems with complicated geometry, Int J numer meth engng, (2002), accepted for publication · Zbl 1072.76048
[11] Golub, G.H.; Ortega, J.M., Scientific computing and differential equations, (1992), Academic Press New York · Zbl 0749.65041
[12] Fasshauer, G.E., Solving partial differential equations by collocation with radial basis functions, (), 131-138 · Zbl 0938.65140
[13] Partridge, P.W.; Brebbia, C.A.; Wrobel, L.W., The dual reciprocity boundary element method, (1992), Computational Mechanics Publication Southampton · Zbl 0758.65071
[14] Chen W, Tanaka M. Relationship between boundary integral equation and radial basis function. The 52nd Symposium of Japan Society for Computational Methods in Engineering (JASCOME) on BEM. Tokyo, September 2000.
[15] Chen, C.S.; Marcozzi, M.; Choi, S., The method of fundamental solutions and compactly supported radial basis functions—a meshless approach to 3D problems, (), 561-570 · Zbl 0972.65098
[16] Itagaki, M., Higher order three-dimensional fundamental solutions to the Helmholtz and the modified Helmholtz equations, Engng anal boundary elements, 15, 289-293, (1995)
[17] Beatson, R.K.; Light, W.A.; Billings, S., Fast solution of the radial basis function interpolation equations: domain decomposition methods, SIAM J sci comput, 22, 5, 1717-1740, (2000) · Zbl 0982.65015
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