Domain Decomposition for Radial Basis Meshless Methods

Numerical Methods for Partial Differential Equations

Volumn 20, Issue 3, 2004, Pages 450-462

  Li, Jichun ^a   Hon, Y C ^b  

^a UNIVERSITY OF NEVADA   (United States)

^b CITY UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Domain decomposition; Meshless method; Radial basis function

Indexed keywords

EID: 2342536532     PISSN: 0749159X     EISSN: None     Source Type: Journal    
DOI: 10.1002/num.10096     Document Type: Article

References (35)

1. 1 M. A. Golberg and C. S. Chen, Discrete projection methods for integral equations, Computational Mechanics Publications, Southampton, 1997.
2. 2 E. J. Kansa, Multiquadric—a scattered data approximation scheme with applications to computational fluid dynamics II, Comput Math Appl 19 (1990), 147–161.
3. 3 Z. Wu, Solving PDE with radial basis functions, Z. Chen, Y. Li, C. A. Micchelli, and Y. Xu, editors, Advances in computational mathematics, Lecture Notes on Pure and Applied Mathematics, Vol. 202, Marcel Dekker, New York, 1999.
4. 4 C. Franke and R. Schaback, Solving partial differential equations by collocation using radial basis functions, Appl Math Comput 93 (1998), 73–82.
5. 5 G. E. Fassauer, Solving partial differential equation by collocation with radial basis functions, A. LeMehaute, C. Rabut, and L. Schumaker, editors, Proceeding of Chamonix, Vanderbilt University Press, 1997, pp. 131–138.
6. 6 M. Griebel and M. A. Schweitzer, A particle‐partition of unity method for the solution of elliptic, parabolic, and hyperbolic PDEs, SIAM J Sci Comput 22 (2000), 853–890.
7. 7 W. K. Liu, Y. Chen, S. Jun, J. S. Chen, T. Belytschko, C. Pan, R. A. Uras, and C. T. Chang, Overview and applications of the reproducing kernel particle methods, Arch Comput Methods Eng: State of the Art Rev 3 (1996), 3–80.
8. 8 T. Belytschko, Y. Y. Lu, and L. Gu, Element‐free Galerkin methods, Int J Numer Meth Engng 37 (1994), 229–256.
9. 9 T. Atluri and T. Zhu, A new meshless local Petrov‐Galerkin (MLPG) approach in computational mechanics, Comput Mech 22 (1998), 117–127.
10. 10 C. A. Duarte and J. T. Oden, H‐p clouds—an h‐p meshless method, Numer Methods Partial Differential Eq 12 (1996), 673–705.
11. 11 I. Babuska and J. M. Melenk, The partition of unity finite element method, Technical Report BN‐1185, Institute for Physical Science and Technology, University of Maryland, 1995.
12. 12 T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl, Meshless methods: an overview and recent developments, Comput Mech Appl Mech Engng 139 (1996), 3–47.
13. 13 M. Zerroukat, H. Power, and C. S. Chen, A numerical method for heat transfer problem using collocation and radial basis functions, Int J Numer Meth Engng 42 (1998), 1263–1278.
14. 14 E. Larsson and B. Fornberg, A numerical study of some radial basis function based solution methods for elliptic PDEs, preprint.
15. 15 J. Li, A. H.‐D. Cheng, and C. S. Chen, On the efficiency and exponential convergence of multiquadric collocation method compared to finite element method, Engng Anal Boundary Elements 27 (2003), 251–257.
16. 16 Y. C. Hon, M. W. Lu, W. M. Xue, and Y. M. Zhu, Multiquadric method for the numerical solution of a biphasic mixture model, Appl Math Comput 88 (1997), 153–175.
17. 17 Y. C. Hon and X. Z. Mao, An efficient numerical scheme for Burgers' equation, Appl Math Comput 95 (1998), 37–50.
18. 18 J. Li, Y. C. Hon, and C. S. Chen, Numerical comparisons of two meshless methods using radial basis functions, Engng Anal Boundary Elements 26 (2002), 205–225.
19. 19 Y. C. Hon, K. F. Cheung, X. Z. Mao, and E. J. Kansa, Multiquadric solution for shallow water equations, ASCE J Hydraulic Engng 125 (1999), 524–533.
20. 20 S. M. Wong, Y. C. Hon, T. S. Li, S. L. Chung, and E. J. Kansa, Multizone decomposition of time‐dependent problems using the multiquadric scheme, Comput Math Appl 37 (1999), 23–43.
21. 21 Y. C. Hon and X. Z. Mao, A radial basis function method for solving options pricing model, Financial Engng 8 (1999), 31–49.
22. 22 B. F. Smith, P. E. Bjorstad, and W. D. Gropp, Domain decomposition: parallel multilevel methods for elliptic partial differential equations, Cambridge University Press, Cambridge, 1996.
23. 23 A. Quarteroni and A. Valli, Domain decomposition methods for partial differential equations, Oxford University Press, Oxford, 1999.
24. 24 P. Le Tallec, Domain decomposition methods in computational mechanics, Comput Mech Adv 1 (1994), 121–220.
25. 25 M. R. Dubal, Domain decomposition and local refinement for multiquadric approximations, Appl Sci Comput 1 (1994), 146–171.
26. 26 R. K. Beatson, W. A. Light, and S. Billings, Fast solution of the radial basis function interpolation equations: domain decomposition methods, SIAM J Sci Comput 22 (2000), 1717–1740.
27. 27 X. Zhou, Y. C. Hon, and J. Li, Overlapping domain decomposition method by radial basis functions, Appl Numer Math 44 (2003), 241–255.
28. 28 E. J. Kansa and Y. C. Hon, Circumventing the ill‐conditioning problem with multiquadric radial basis functions: application to elliptic partial differential equations, Comput Math Appl 39 (2000), 123–137.
29. 29 M. J. D. Powell, The theory of radial basis function approximation in 1990, W. Light, editor, Advances in numerical analysis, Vol. II, Oxford Science Publications, Oxford, 1992, pp. 105–210.
30. 30 M. S. Ingber, C. S. Chen, J. A. Tanski, A mesh free approach using radial basis functions and parallel domain decomposition for solving three dimensional diffusion equations, Int J Numer Meth Engrg (in press).
31. 31 L. Ling and E. J. Kansa, Preconditioning for radial basis functions with domain decomposition methods, International Workshop on MeshFree Methods 2003, Instituto Superior Tecnico Lisbon, Portugal, July 21–23, 2003.
32. 32 S. N. Atluri and S. Shen, The meshless local Petrov‐Galerkin (MLPG) method, Tech Science Press, Encino, CA, 2002.
33. 33 G. R. Liu, Mesh free methods, CRC Press, Boca Raton, FL, 2002.
34. 34 M. Griebel and M. A. Schweitzer, editors, Meshfree methods for partial differential equations, Springer, New York, 2003.
35. 35 M. D. Buhmann, Radial basis functions, Cambridge University Press, Cambridge, 2003.