A note on the meshless method using radial basis functions

Computers and Mathematics with Applications

Volumn 55, Issue 1, 2008, Pages 66-75

  Duan, Yong ^a  

^a UNIVERSITY OF ELECTRONIC SCIENCE AND TECHNOLOGY OF CHINA   (China)

Author keywords

Collocation method; Galerkin method; Inverse inequalities; Meshless; Radial basis function

Indexed keywords

GALERKIN METHODS; NUMERICAL METHODS; PROBLEM SOLVING; RADIAL BASIS FUNCTION NETWORKS;

COLLOCATION METHODS; INVERSE INEQUALITIES; MESHLESS METHODS;

INVERSE PROBLEMS;

EID: 36349022096     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2007.03.011     Document Type: Article

References (41)

1. Griebel M., and Schweitzer M.A. A particle-partition of unity method for the solution of elliptic, parabolic and hyperbolic PDEs. SIAM J. Sci. Comput. 22 (2000) 853-890
2. Babuška I., Banerjee U., and Osborn J.E. Survey of meshless and generalized finite element methods: A unified approach. Acta. Numer (2003), Cambridge University Press 1-125
3. Belytschko T., Krongauze Y., Organ D., Fleming M., and Krysl P. Meshless methods: An overview and recent developments. Comput. Methods. Appl. Math. 139 (1996) 3-47 (special issue on Meshless Methods)
4. Kansa E.J. Multiquadrics-A scattered data approximation scheme with applications to computational fluid dynamics, II. Solution to parabolic, hyperbolic and elliptic partial differential equations. Comput. Math. Appl. 19 (1990) 147-161
5. Liu G.R. Mesh Free Methods: Moving Beyond the Finite Element Method (2003), CRC Press
6. Narcowich F.J., and Ward J.D. Scattered data interpolation on spheres: Error estimates and locally supported basis functions. SIAM J. Math. Anal. 33 (2002) 1393-1410
7. Schaback R. Error estimates and condition numbers for radial basis function interpolation. Adv. Comput. Math. 3 (1995) 251-264
8. Schaback R. Improved error bounds for scattered data interpolation by radial basis functions. Math. Comp. 68 (1999) 201-216
9. Wendland H. Error estimates for interpolation by compactly supported radial basis functions of minimal degree. J. Approx. Theory 93 (1998) 258-272
10. Wendland H. Scattered Data Approximation (2005), Cambridge University Press
11. Buhmann M.D. Radial Basis Functions: Theory and Implementations (2003), Cambridge University Press
12. Dubal M.R. Domain decomposition and local refinement for multiquadric approximation. I. Second order equations in one-dimension. Appl. Sci. Comput. 1 (1994) 146-171
13. Fasshauer G.E. Solving differential equations with radial basis functions: Multilevel methods and smoothing. Adv. Comput. Math. 11 (1999) 139-159
14. Fedoseyev A.I., Friedman M.J., and Kansa E.J. Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary. Radial basis functions and partial differential equations. Comput. Math. Appl 43 (2002) 439-455
15. Franke C., and Schaback R. Convergence order estimates of meshless collocation methods using radial basis functions. Adv. Comput. Math. 8 (1998) 381-399
16. Golberg M.A., Chen C.S., and Karur S.R. Improved multiqudric approximation for partial differential equations. Eng. Anal. Bound. Elem. 18 (1996) 9-17
17. Wendland H. Meshless Galerkin methods using radial basis functions. Math. Comp. 68 (1999) 1521-1531
18. Hon Y.C., and Schaback R. On unsymmetric collocation by radial basis functions. Appl. Math. Comput. 119 (2001) 177-186
19. Babuška I. The finite element method with penalty. Math. Comp. 27 (1973) 221-228
20. Nistche J.A. Über ein Variationsprinzip zur Lösung Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unteworfen sind. Abh. Math. Sem. Univ. Hamburg 36 (1971) 9-15
21. Wendland H. Sobolev-type error estimates for interpolation by radial basis functions. In: LeMéhauté A., Rabut C., and Schumaker L.L. (Eds). Surface Fitting and Multiresolution Methods (1997), Vanderbilt University Press, Nashville 337-344
22. Wu Z. Compactly supported positive definite radial functions. Adv. Comput. Math. 4 (1995) 283-292
23. Adams R.A. Sobolev Spaces (1975), Academic Press
24. Hon Y.C., and Wu Z. Additive schwarz domain decomposition with radial basis approximation. Int. J. Appl. Math. 4 (2000) 81-98
25. Ciarlet P.G. The Finite Element Method for Elliptic Problems (1978), Amsterdam, North-Holland
26. Grisvard P. Elliptic Problems in Nonsmooth Domains (1985), Pitman
27. Babuška I. The finite element method with Lagrange multipliers. Numer. Math. 20 (1973) 179-192
28. Brezzi F. On the existence,uniqueness and approximations of saddlepoint problems arising from Lagrange multipliers. RAIRO Anal. Numèr. 8 (1974) 129-151
29. Stenberg R. On some techniques for approximating boundary conditions in the finite element methods. J. Comput. Appl. Math. 63 (1995) 139-148
30. Duan Y., and Tan Y.J. A meshless Galerkin method for Dirichlet problem using radial basis functions. J. Comput. Appl. Math. 196 (2006) 394-401
31. Barbosa H.J.C., and Hughes T.J.R. Boundary Lagrange multipliers in finite element methods: Error analysis in natural norms. Numer. Math. 62 (1992) 1-15
32. Lions J.L. Problèmes aux limites non homogènes à donées irrégulières: Une méthode d'approximation. Numerical Analysis of Partial Differential Equations (C.I.M.E.2 Ciclo, Ispra, 1967) (1968), Edizioni Cremonese, Rome 283-292
33. Beatson R.K., Light W.A., and Billings S. Fast solution of the radial basis function interpolation equations: Domain decomposition methods. SIAM J. Sci. Comput. 22 (2000) 1717-1740
34. Duan Y., and Tan Y.J. Meshless Galerkin method based on regions partitioned into subdomains using global RBF. Proceedings of ICNAAM 2004 (2004), Wiley-VCH 128-133
35. Ling L., and Kansa E.J. Preconditioning for radial basis functions with domain decomposition methods. Math. Comput. Modelling 40 (2004) 1413-1427
36. Ling L., and Hon Y.C. Improved numerical solver for Kansa's method based on affine space decomposition. Eng. Anal. Bound. Elem. 29 (2005) 1077-1085
37. Kansa E.J., and Hon Y.C. Circumventing the Ill-conditioning problem with multiquadric radial basis functions: Applications for elliptic partial differential equations. Comput. Math. Appl. 39 (2000) 123-137
38. Ling L., Opfer R., and Schaback R. Results on meshless collocation techniques. Eng. Anal. Bound. Elem. 30 (2006) 247-253
39. Ling L., and Kansa E.J. A least squares preconditioner for radial basis functions collocation methods. Adv. Comput. Math. 23 (2005) 31-54
40. Li J., and Hon Y.C. Domain decomposition for radial basis meshless methods. Numer. Methods Partial Differential Equations 20 (2004) 450-462
41. Zhou X., Hon Y.C., and Li J. Overlapping domain decomposition method by radial basis functions. Appl. Numer. Math. 44 (2003) 241-255