## A meshless, integration-free, and boundary-only RBF technique.(English)Zbl 0999.65142

Summary: Based on the radial basis function (RBF), nonsingular general solution, and dual reciprocity method (DRM), this paper presents an inherently meshless, integration-free, boundary-only RBF collocation technique for numerical solution of various partial differential equation systems. The basic ideas behind this methodology are mathematically very simple.
In this study, the RBFs are employed to approximate the inhomogeneous terms via the DRM, while nonsingular general solution leads to a boundary-only RBF formulation for homogeneous solution. The present scheme is named as the boundary knot method (BKM) to differentiate it from the other numerical techniques. In particular, due to the use of nonsingular general solutions rather than singular fundamental solutions, the BKM is different from the method of fundamental solution in that the former does not require the artificial boundary and results in the symmetric system equations under certain conditions.
The efficiency and utility of this new technique are validated through a number of typical numerical examples. Completeness concern of the BKM due to the sole use of the nonsingular part of complete fundamental solution is also discussed.

### MSC:

 65N38 Boundary element methods for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
Full Text:

### References:

 [1] Nardini, D.; Brebbia, C. A., A new approach to free vibration analysis using boundary elements, Applied Mathematical Modeling, 7, 157-162 (1983) · Zbl 0545.73078 [2] Golberg, M. A.; Chen, C. S., The method of fundamental solutions for potential, Helmholtz and diffusion problems, (Golberg, M. A., Boundary Integral Methods—Numerical and Mathematical Aspects (1998), Computational Mechanics Publications), 103-176 · Zbl 0945.65130 [3] Chen, C. S., The method of potential for nonlinear thermal explosion, Commun. Numer. Methods Engng., 11, 675-681 (1995) · Zbl 0839.65143 [4] Golberg, M. A.; Chen, C. S.; Bowman, H.; Power, H., Some comments on the use of radial basis functions in the dual reciprocity method, Comput. Mech., 21, 141-148 (1998) · Zbl 0931.65116 [5] Muleskov, A. S.; Golberg, M. A.; Chen, C. S., Particular solutions of Helmholtz-type operators using higher order polyharmonic splines, Comput. Mech., 23, 411-419 (1999) · Zbl 0938.65139 [6] Kitagawa, T., On the numerical stability of the method of fundamental solutions applied to the Dirichlet problem, Japan Journal of Applied Mathematics, 35, 507-518 (1988) [7] Kitagawa, T., Asymptotic stability of the fundamental solution method, Journal of Computational and Applied Mathematics, 38, 263-269 (1991) · Zbl 0752.65077 [8] Kansa, E. J., Multiquadrics: A scattered data approximation scheme with applications to computational fluid-dynamics, Computers Math. Applic., 19, 8/9, 147-161 (1990) · Zbl 0850.76048 [9] Kansa, E. J.; Hon, Y. C., Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations, Computers Math. Applic., 39, 7/8, 123-137 (2000) · Zbl 0955.65086 [10] Hon, Y. C.; Mao, X. Z., A radial basis function method for solving options pricing model, Financial Engineering, 81, 1, 31-49 (1999) [11] Kamiya, N.; Andoh, E., A note on multiple reciprocity integral formulation for the Helmholtz equation, Commun. Numer. Methods Engng., 9, 9-13 (1993) · Zbl 0781.65093 [12] Chen, J. T.; Huang, C. X.; Chen, K. H., Determination of spurious eigenvalues and multiplicities of true eigenvalues using the real-part dual BEM, Comput. Mech., 24, 41-51 (1999) · Zbl 0951.76051 [13] Power, H., On the completeness of the multiple reciprocity series approximation, Commun. Numer. Methods Engng., 11, 665-674 (1995) · Zbl 0831.76052 [14] Chen, W.; Tanaka, M., New advances in dual reciprocity and boundary-only RBF methods, (Tanaka, M., Proceeding of BEM Technique Conference, 10 (2000)), 17-22, Tokyo, Japan [15] Katsikadelis, J. T.; Nerantzaki, M. S., The boundary element method for nonlinear problems, Engineering Analysis with Boundary Element, 23, 273-365 (1999) · Zbl 0945.65132 [16] Zerroukat, M.; Power, H.; Chen, C. S., A numerical method for heat transfer problems using collocation and radial basis function, Inter. J. Numer. Method Engng., 42, 1263-1278 (1998) · Zbl 0907.65095 [17] Wrobel, L. C.; DeFigueiredo, D. B., A dual reciprocity boundary element formulation for convection-diffusion problems with variable velocity fields, Engng. Analysis with BEM, 8, 6, 312-319 (1991) [18] Schclar, N. A., Anisotropic Analysis Using Boundary Elements (1994), Comput. Mech. Publ: Comput. Mech. Publ Southampton · Zbl 0996.74505 [19] Kogl, M.; Gaul, L., Dual reciprocity boundary element method for three-dimensional problems of dynamic piezoelectricity, (Boundary Elements XXI (1999), Comput. Mech. Publ: Comput. Mech. Publ Southampton), 537-548 · Zbl 1052.74600 [20] Partridge, P. W.; Brebbia, C. A.; Wrobel, L. W., The Dual Reciprocity Boundary Element Method (1992), Comput. Mech. Publ: Comput. Mech. Publ Southampton, U.K · Zbl 0758.65071 [21] Wendland, H., Piecewise polynomial, positive definite and compactly supported radial function of minimal degree, Adv. Comput. Math., 4, 389-396 (1995) · Zbl 0838.41014 [22] Wu, Z., Multivariate compactly supported positive definite radial functions, Adv. Comput. Math., 4, 283-292 (1995) · Zbl 0837.41016 [23] Schaback, R., Creating surfaces from scattered data using radial basis function, (Dahlen, M., Mathematical Methods for Curves and Surfaces (1995), Vanderbilt University Press: Vanderbilt University Press Nashville, TN), 477-496 · Zbl 0835.65036 [25] Chen, C. S.; Brebbia, C. A.; Power, H., Boundary element methods using compactly supported radial basis functions, Commun. Numer. Meth. Engng., 15, 137-150 (1999) · Zbl 0927.65140 [26] Hardy, R. L., Multiquadratic equations for topography and other irregular surfaces, J. Geophys. Res., 176, 1905-1915 (1971) [27] Duchon, J., Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces, RAIRO Analyse Numeriques, 10, 5-12 (1976) [28] Franke, R., Scattered data interpolation: Tests of some methods, Math. Comput., 48, 181-200 (1982) · Zbl 0476.65005 [29] Hon, Y. C.; Mao, X. Z., An efficient numerical scheme for Burgers’ equation, Appl. Math. Comput., 95, 1, 37-50 (1998) · Zbl 0943.65101 [30] Carlson, R. E.; Foley, T. A., The parameter $$R^2$$ in multiquadratic interpolation, Computers Math. Applic., 21, 9, 29-42 (1991) · Zbl 0725.65009 [31] Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P., Numerical Recipes in Fortran (1992), Cambridge University Press · Zbl 0778.65002 [32] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Methods in Fluid Dynamics (1988), Springer: Springer Berlin · Zbl 0658.76001 [33] Dai, D. N., An improved boundary element formulation for wave propagation problems, Engng. Analysis with Boundary Element, 10, 277-281 (1992) [34] Piltner, R., Recent development in the Trefftz method for finite element and boundary element application, Advances in Engineering Software, 2, 107-115 (1995) · Zbl 0984.65504 [35] Duchon, J., Splines minimizing rotation invariant semi-norms in Sobolev spaces, Constructive Theory of Functions of Several Variables (1976), Springer-Verlag: Springer-Verlag Berlin
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.