## Meshless Galerkin methods using radial basis functions.(English)Zbl 1020.65084

Summary: We combine the theory of radial basis functions with the field of Galerkin methods to solve partial differential equations (PDE). After a general description of the method we show convergence and derive error estimates for smooth problems in arbitrary dimensions. We restrict ourselves to second-order PDE, but a generalization to higher-order equations can be done in an obvious way.

### MSC:

 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 41A30 Approximation by other special function classes

### Keywords:

approximation orders; positive definite functions; PDE
Full Text:

### References:

 [1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. · Zbl 0314.46030 [2] T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, P. Krysl, Meshless methods: an overview and recent developments, Computer Methods in Applied Mechanics and Engineering, special issue on Meshless Methods, vol 139 (1996), pp 3- 47. · Zbl 0891.73075 [3] Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. · Zbl 0804.65101 [4] Julio G. Dix and Robert D. Ogden, An interpolation scheme with radial basis in Sobolev spaces \?^{\?}(\?$$^{n}$$), Rocky Mountain J. Math. 24 (1994), no. 4, 1319 – 1337. · Zbl 0819.41004 [5] Jean Duchon, Sur l’erreur d’interpolation des fonctions de plusieurs variables par les \?^{\?}-splines, RAIRO Anal. Numér. 12 (1978), no. 4, 325 – 334, vi (French, with English summary). · Zbl 0403.41003 [6] N. Dyn, Interpolation and approximation by radial and related functions, Approximation theory VI, Vol. I (College Station, TX, 1989) Academic Press, Boston, MA, 1989, pp. 211 – 234. · Zbl 0705.41006 [7] W. R. Madych and S. A. Nelson, Multivariate interpolation and conditionally positive definite functions, Approx. Theory Appl. 4 (1988), no. 4, 77 – 89. · Zbl 0703.41008 [8] Will Light , Advances in numerical analysis. Vol. II, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992. Wavelets, subdivision algorithms, and radial basis functions. · Zbl 0744.00036 [9] Eduard Prugovečki, Quantum mechanics in Hilbert space, Academic Press, New York-London, 1971. Pure and Applied Mathematics, Vol. 41. · Zbl 0217.44204 [10] R. Schaback, Creating surfaces from scattered data using radial basis functions, Mathematical methods for curves and surfaces (Ulvik, 1994) Vanderbilt Univ. Press, Nashville, TN, 1995, pp. 477 – 496. · Zbl 0835.65036 [11] Robert Schaback, Multivariate interpolation and approximation by translates of a basis function, Approximation theory VIII, Vol. 1 (College Station, TX, 1995) Ser. Approx. Decompos., vol. 6, World Sci. Publ., River Edge, NJ, 1995, pp. 491 – 514. · Zbl 1139.41301 [12] R. Schaback, Approximation by radial basis functions with finitely many centers, Constr. Approx. 12 (1996), no. 3, 331 – 340. · Zbl 0855.41011 [13] Robert Schaback and Z. Wu, Operators on radial functions, J. Comput. Appl. Math. 73 (1996), no. 1-2, 257 – 270. · Zbl 0857.42004 [14] Holger Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math. 4 (1995), no. 4, 389 – 396. · Zbl 0838.41014 [15] H. Wendland, Sobolev-type error estimates for interpolation by radial basis functions, in: Surface Fitting and Multiresolution Methods, A. LeMéhauté, C. Rabut, L. L. Schumaker, eds., Vanderbilt University Press, Nashville, 1997, pp 337-344. · Zbl 0955.41002 [16] H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, Journal of Approx. Theory 93 (1998), pp 258-272. CMP 98:11 [17] Zong Min Wu and Robert Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal. 13 (1993), no. 1, 13 – 27. · Zbl 0762.41006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.