On the increasingly flat radial basis function and optimal shape parameter for the solution of elliptic PDEs

Engineering Analysis with Boundary Elements

Volumn 34, Issue 9, 2010, Pages 802-809

  Huang, C S ^a   Yen, H D ^a   Cheng, A H D ^b  

^a NATIONAL SUN YAT SEN UNIVERSITY   (Taiwan)

^b The University of Mississippi   (United States)

Author keywords

Arbitrary precision computation; Error estimate; Increasingly flat radial basis function; Meshless method; Multiquadric collocation method

Indexed keywords

FUNCTIONS; INTERPOLATION; LAGRANGE MULTIPLIERS; RADIAL BASIS FUNCTION NETWORKS;

ARBITRARY PRECISION; COLLOCATION METHOD; ERROR ESTIMATES; MESH-LESS METHODS; RADIAL BASIS FUNCTIONS;

POLYNOMIALS;

EID: 79957742226     PISSN: 09557997     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.enganabound.2010.03.002     Document Type: Article

References (29)

1. Franke R. Scattered data interpolation-tests of some methods. Math Comput 1982;38:181-200.
2. Hardy RL. Multiquadric equations of topography and other irregular surfaces. J Geophys Res 1971;76:1905-15.
3. Hardy RL. Theory and applications of the multiquadric-biharmonic method-20 years of discovery. Comput Math Appl 1990;19:163-208. (Pubitemid 20668366)
4. Kansa EJ. Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics. 1. Surface approximations and partial derivative estimates. Comput Math Appl 1990;19:127-45. (Pubitemid 20668364)
5. Kansa EJ. Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics. 2. Solutions to parabolic, hyperbolic and elliptic partial-differential equations. Comput Math Appl 1990;19:147-61. (Pubitemid 20668365)
6. Hon YC, Cheung KF, Mao XZ, Kansa EJ. Multiquadric solution for shallow water equations. J Hydraul Eng ASCE 1999;125:524-33. (Pubitemid 29400985)
7. Hon YC, Lu MW, Xue WM, Zhu YM. Multiquadric method for the numerical solution of a biphasic mixture model. Appl Math Comput 1997;88:153-75. (Pubitemid 127683562)
8. Vertnik R, Sarler B. Meshless local radial basis function collocation method for convective-diffusive solid-liquid phase change problems. Int J Numer Methods Heat & Fluid Flow 2006;16:617-40. (Pubitemid 43883158)
9. Wen PH, Hon YC. Geometrically nonlinear analysis of Reissner-Mindlin plate by meshless computation. Comput Modeling Eng & Sci 2007;21:177-91. (Pubitemid 47598983)
10. Cheng AH-D, Cabral JJSP. Direct solution of ill-posed boundary value problems by radial basis function collocation method. Int J Numer Methods Eng 2005;64:45-64. (Pubitemid 41300194)
11. Tarwater AE. Parameter study of Hardy's multiquadric method for scattered data interpolation. Technical Report UCRL-54670; 1985, 69p.
12. Madych WR. Miscellaneous error-bounds for multiquadric and related interpolators. Comput Math Appl 1992;24:121-38.
13. Cheng AH-D, Golberg MA, Kansa EJ, Zammito G. Exponential convergence and h-c multiquadric collocation method for partial differential equations. Numer Methods Partial Diff Eqns 2003;19:571-94.
14. Huang C-S, Lee C-F, Cheng AH-D. Error estimate, optimal shape factor, and high precision computation of multiquadric collocation method. Eng Anal Boundary Elem 2007;31:614-23. (Pubitemid 46843545)
15. 1, /2 as c-oo. Comput Math Appl 1992;24:1-6.
16. Driscoll TA, Fornberg B. Interpolation in the limit of increasingly flat radial basis functions. Comput Math Appl 2002;43:413-22. (Pubitemid 34104977)
17. Fornberg B, Wright G, Larsson E. Some observations regarding interpolants in the limit of flat radial basis functions. Comput Math Appl 2004;47:37-55.
18. Larsson E, Fornberg B. Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions. Comput Math Appl 2005;49:103-30. (Pubitemid 40597177)
19. Schaback R. Multivariate interpolation by polynomials and radial basis functions. Constructive Approximation 2005;21:293-317. (Pubitemid 40900286)
20. Fairweather G, Karageorghis A. The method of fundamental solutions for elliptic boundary value problems. Adv Comput Math 1998;9:69-95. (Pubitemid 128403959)
21. Bogomolny A. Fundamental-solutions method for elliptic boundary-value problems. SIAM J Numer Anal 1985;22:644-69. (Pubitemid 15528321)
22. Chen JT, Lee YT, Yu SR, Shieh SC. Equivalence between the Trefftz method and the method of fundamental solution for the annular Green's function using the addition theorem and image concept. Eng Anal Boundary Elem 2009;33: 678-88.
23. Schaback R. Adaptive numerical solution of MFS systems. In: Chen CS, Karageorghis A, Smyrlis YS, editors. The method of fundamental solutions-a meshless method. Atlanta, GA, Dynamic Publishers; 2008. p. 1-27. [chapter 1].
24. Li Z-C, Lu Z-Z, Hu H-Y, Cheng AH-D. Trefftz and collocation methods. Southampton, Boston, WIT Press; 2008.
25. Kolodziej JA, Zielinski AP. Boundary collocation techniques and their application in engineering. Southampton, Boston, WIT Press; 2009.
26. Buhmann MD. Radial basis functions: theory and implementations. Cambridge, UK, Cambridge University Press; 2003.
27. Wendland H. Scattered data approximation. Cambridge, UK, Cambridge University Press; 2005.
28. Micchelli CA. Interpolation of scattered data-distance matrices and conditionally positive definite functions. Constructive Approximation 1986;2:11-22.
29. Yen H-D. On the increasingly flat RBFs based solution methods for elliptic PDEs and interpolations. Master thesis, Department of Applied Mathematics, National Sun Yat-sen University, Taiwan; 2008, 70p.