Stable and convergent unsymmetric meshless collocation methods

SIAM Journal on Numerical Analysis

Volumn 46, Issue 3, 2008, Pages 1097-1115

  Ling, Leevan ^a   Schaback, Robert ^b  

^a HONG KONG BAPTIST UNIVERSITY   (Hong Kong)

^b UNIVERSITY OF GÖTTINGEN   (Germany)

Author keywords

Convergence; Error bounds; Kansa's method; Linear optimization

Indexed keywords

ADAPTIVE ALGORITHMS; OPTIMIZATION;

BACKGROUND THEORIES; COLLOCATION METHODS; CONVERGENCE; ERROR BOUNDS; KANSA'S METHOD; LINEAR OPTIMIZATION; LINEAR OPTIMIZATIONS; MESHLESS; NUMERICAL EXPERIMENTS; NUMERICAL IMPLEMENTATIONS; TRIAL SPACES;

CONVERGENCE OF NUMERICAL METHODS;

EID: 55349130353     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/06067300X     Document Type: Article

References (17)

1. S. N. ATLURI AND S. P. SHEN, The Meshlesa Local Petrov-Galerkin (MLPG) Method, Tech Science Press, Encino, CA, 2002.
2. ∞, linear approximations on a discrete set, Numer. Math., 8 (1966), pp. 295-306.
3. C. FRANKE AND R. SCHABACK, Convergence order estimates of meshless collocation methods using radial basis functions, Adv. Comput. Math., 8 (1998), pp. 381-399.
4. C. FRANKE AND R. SCHABACK, Solving partial differential equations by collocation using radial basis functions, Appl. Math. Comput., 93 (1998), pp. 73-82.
5. M. A. GOLBERG AND C. S. CHEN, EDS., The dual reciprocity method and radial basis functions, Eng. Anal. Boundary Elements, 24 (2000), pp. 7-8.
6. Y. C. HON AND R. SCHABACK, On unsymmetric collocation by radial basis functions, Appl. Math. Comput., 119 (2001), pp. 177-186.
7. E. J. KANSA AND Y. C. HON, Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations, Comput. Math. Appl., 39 (2000), pp. 123-137.
8. 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 (1990), pp. 127-145.
9. 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 (1990), pp. 147-161.
10. O. A. LADYZHENSKAYA, The boundary value problems of mathematical physics, Appl. Math. Sci. 49, Springer-Verlag, New York, 1985 (translated from Russian by Jack Lohwater [Arthur J. Lohwater]).
11. L. LING, R. OPFER, AND R. SCHABACK, Results on meshless collocation techniques, Eng. Anal. Boundary Elements, 30 (2006), pp. 247-253.
12. W. R. MADYCH, Miscellaneous error bounds for multiquadric and related interpolators, Comput. Math. Appl., 24 (1992), pp. 121-138.
13. R. SCHABACK, On the versatility of meshless kernel methods, in Advances in Computational & Experimental Engineering & Sciences 428, S.N. Atluri, D.E. Beskos, and D. Polyzos, eds., Universität Göttingen, 2003.
14. R. SCHABACK, Convergence of unsymmetric kernel-based meshless collocation methods, SIAM J. Numer. Anal., 45 (2005), pp. 333-351.
15. R. SCHABACK, Unsymmetric Meshless Methods for Operator Equations, 2006, http://www.num.math.uni-goettingen.de/schaback.
16. H. WENDLAND, Scattered Data Approximation, Cambridge University Press, London, 2005.
17. Z. M. WU, Hermite-Birkhoff interpolation of scattered data by radial basis functions, Approx. Theory Appl., 8 (1992), pp. 1-10.