On convergence of a least-squares kansa's method for the modified helmholtz equations

Advances in Applied Mathematics and Mechanics

Volumn 1, Issue 3, 2009, Pages 367-382

  Kwok, Ting On ^a   Ling, Leevan ^a  

^a HONG KONG BAPTIST UNIVERSITY   (Hong Kong)

Author keywords

Adaptive greedy algorithm; Asymmetric collocation; Convergence analysis; Kansa's method; Radial basis function

Indexed keywords

EID: 71949128335     PISSN: 20700733     EISSN: 20751354     Source Type: Journal    
DOI: None     Document Type: Article

References (24)

1. D. BROWN, L. LING, E. J. KANSA AND J. LEVESLEY, On approximate cardinal preconditioning methods for solving PDEs with radial basis functions, Eng. Anal. Bound. Elem., 29 (4) (2005), pp. 343-353.
2. M. D. BUHMANN, Radial Basis Functions: Theory and Implementations, volume 12, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2003.
3. H. DING, C. SHU AND D.B. TANG, Error estimates of local multiquadric-based differential quadrature (LMQDQ) method through numerical experiments, Int. J. Numer. Methods Eng., 63(11) (2005), pp. 1513-1529.
4. G. E. FASSHAUER, Meshfree Approximation Methods with Matlab, Interdisciplinary Mathematical Sciences 6. Hackensack, NJ: World Scientific., 2007.
5. Y. C. HON AND ROBERT SCHABACK, On unsymmetric collocation by radial basis functions, Appl. Math. Comput., 119(2-3) (2001), pp. 177-186.
6. E. J. KANSA, Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics. I. Surface approximations and partial derivative estimates, Comput. Math. Appl., 19(8-9) (1990), pp. 127-145.
7. E. J. KANSA, Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics. II. Solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput. Math. Appl., 19(8-9) (1990), pp. 147-161.
8. C.-F. LEE, L. LING AND R. SCHABACK, On convergent numerical algorithms for unsymmetric collocation, To appear in Adv. Comput. Math., (2007).
9. L. LING AND Y. C. HON, Improved numerical solver for Kansa's method based on affine space decomposition, Eng. Anal. Bound. Elem., 29(12) (2005), pp. 1077-1085.
10. L. LING AND E. J. KANSA, Preconditioning for radial basis functions with domain decomposition methods, Math. Comput. Modelling, 40(13) (2004), pp. 1413-1427.
11. L. LING, R. OPFER AND R. SCHABACK, Results on meshless collocation techniques, Eng. Anal. Bound. Elem., 30(4) (2006), pp. 247-253.
12. L. LING AND ROBERT SCHABACK, An improved subspace selection algorithm for meshless collocation methods, In review, 2008.
13. L. LING AND R. SCHABACK, Stable and convergent unsymmetric meshless collocation methods, SIAM J. Numer. Anal., 46(3) (2008), pp. 1097-1115.
14. W. R. MADYCH, Miscellaneous error bounds for multiquadric and related interpolators, Comput. Math. Appl., 24(12) (1992), pp. 121-138.
15. W. R. MADYCH AND S. A. NELSON, Multivariate interpolation and conditionally positive definite functions, Approx. Theory Appl., 4(4) (1988), pp. 77-89.
16. W.R. MADYCH, Error estimates for interpolation by generalized splines, Curves and Surfaces, Pap. Int. Conf., Chamonix-Mont-Blanc/Fr. 1990, 297-306 (1991).
17. G. J. MORIDIS AND E. J. KANSA, The Laplace transform multiquadrics method: a highly accurate scheme for the numerical solution of linear partial differential equations, J. Appl. Sci. Comput., 1(2) (1994), pp. 375-407.
18. F. J. NARCOWICH, J. D. WARD AND H. WENDLAND, Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting, Math. Comp., 74(250) (2005), pp. 743-763.
19. R. SCHABACK, Error estimates and condition numbers for radial basis function interpolation, Adv. Comput. Math., 3(3) (1995), pp. 251-264.
20. R. SCHABACK, Approximation by radial basis functions with finitely many centers, Constr. Approx., 12(3) (1996), pp. 331-340.
21. R. SCHABACK, Convergence of unsymmetric kernel-based meshless collocation methods, SIAM J. Numer. Anal., 45(1) (2007), pp. 333-351.
22. H. WENDLAND, Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics (No. 17). Cambridge University Press, Cambridge, 2005.
23. J. WERTZ, E. J. KANSA AND L. LING, The role of the multiquadric shape parameters in solving elliptic partial differential equations, Comput. Math. Appl., 51(8) (2006), pp. 1335-1348.
24. X. ZHOU, Y.C. HON AND K.F. CHEUNG, A grid-free, nonlinear shallow-water model with moving boundary, Eng. Anal. Bound. Elem., 28(9) (2004), pp. 1135-1147.