×

zbMATH — the first resource for mathematics

Multilevel compact radial functions based computational schemes for some elliptic problems. (English) Zbl 0999.65143
Summary: Compactly supported radial basis functions (CS-RBFs) have been recently introduced in the context of the dual reciprocity method as a possible curve of dense matrices and ill-conditioning problems when using the classical radial basis functions. However, the support scaling factor and slow convergence rate of the CS-RBFs have also raised issues on the effectiveness of the CS-RBFs. In this paper, two multilevel schemes have been proposed to alleviate these problems.

MSC:
65N38 Boundary element methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Kansa, E., Multiquadrics: A scattered data approximation scheme with applications to computational fluid-dynamics, Computers math. applic., 19, 8/9, 147-161, (1990) · Zbl 0850.76048
[2] Partridge, P.W.; Brebbia, C.A.; Wrobel, L.C., The dual reciprocity boundary element method, (1992), Computational Mechanics · Zbl 0758.65071
[3] Cheng, A.H.-D.; Young, D.-L.; Tsai, J.-J., Solution of Poisson’s equation by iterative DRBEM using compactly-supported, positive-definite radial basis function, Eng. analy. boundary elements, 24, 549-557, (2000) · Zbl 0966.65089
[4] Schaback, R., On the efficiency of interpolation by radial basis functions, (), 309-318 · Zbl 0937.65013
[5] Dubal, M.R., Domain decomposition and local refinement for multiquadric approximations I: second-order equation in one-dimension, J. appl. sci. comp., 1, 146-171, (1994)
[6] Popov, V.; Power, H., The DRM-MD integral equation method: an efficient approach for the numerical solution of domain dominant problems, Int. J. numer. methods eng., 44, 327-353, (1999) · Zbl 0946.76052
[7] Schaback, R., Creating surfaces from scattered data using radial basis functions, (), 477-496 · Zbl 0835.65036
[8] Wendland, H., Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. comput. math., 4, 389-396, (1995) · Zbl 0838.41014
[9] Wu, Z., Multivariate compactly supported positive definite radial functions, Adv. comput. math., 4, 283-292, (1995) · Zbl 0837.41016
[10] Chen, C.S.; Brebbia, C.A.; Power, H., The dual reciprocity method using compactly supported radial basis functions, Comm. num. meth. eng., 15, 137-150, (1999) · Zbl 0927.65140
[11] Chen, C.S.; Marcozzi, M.; Choi, S., The method of fundamental solutions and compactly supported radial basis functions—A meshless approach in 3D problems, (), 313-322
[12] Fasshauer, G.E., Solving differential equations with radial basis functions: multilevel methods and smoothing, Adv. in comp. math., 11, 139-159, (1999) · Zbl 0940.65122
[13] Golberg, M.A.; Chen, C.S.; Ganesh, M., Particular solutions of 3D Helmholtz-type equations using compactly supported radial basis functions, Engineering analysis with boundary elements, 24, 539-547, (2000) · Zbl 0994.76058
[14] Floater, M.S.; Iske, A., Multistep scattered data interpolation using compactly supported radial basis functions, J. comp. appl. math., 73, 65-78, (1996) · Zbl 0859.65006
[15] Golberg, M.A.; Chen, C.S., Discrete projection methods for integral equations, (1986), Computational Mechanics Southampton
[16] Bogomolny, A., Fundamental solutions method for elliptic boundary value problems, SIAM J. numer. anal., 22, 644-669, (1985) · Zbl 0579.65121
[17] Fairweather, G.; Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Advances in comp. math., 9, 69-95, (1998) · Zbl 0922.65074
[18] Golberg, M.A.; Chen, C.S., The method of fundamental solutions for potential, Helmholtz and diffusion problems, (), 105-176, Chapter 4 · Zbl 0945.65130
[19] Niederreiter, H., Random number generation and quasi-Monte Carlo methods, (1992), SIAM, CBMS 63 Philadelphia, PA · Zbl 0761.65002
[20] Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P., Numerical recipes in Fortran: the art of scientific computing, (1996), Cambridge University Press Cambridge · Zbl 0892.65001
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.