Meshless Galerkin methods using radial basis functions
Tóm tắt
We combine the theory of radial basis functions with the field of Galerkin methods to solve partial differential equations. After a general description of the method we show convergence and derive error estimates for smooth problems in arbitrary dimensions.
Từ khóa
Tài liệu tham khảo
Adams, Robert A., 1975, Sobolev spaces
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.
Brenner, Susanne C., 1994, The mathematical theory of finite element methods, 15, 10.1007/978-1-4757-4338-8
Dix, Julio G., 1994, An interpolation scheme with radial basis in Sobolev spaces 𝐻^{𝑠}(𝑅ⁿ), Rocky Mountain J. Math., 24, 1319, 10.1216/rmjm/1181072340
Duchon, Jean, 1978, Sur l’erreur d’interpolation des fonctions de plusieurs variables par les 𝐷^{𝑚}-splines, RAIRO Anal. Num\'{e}r., 12, 325, 10.1051/m2an/1978120403251
Dyn, N., 1989, Interpolation and approximation by radial and related functions, 211
Madych, W. R., 1988, Multivariate interpolation and conditionally positive definite functions, Approx. Theory Appl., 4, 77
1992, Advances in numerical analysis. Vol. II
Prugovečki, Eduard, 1971, Quantum mechanics in Hilbert space
Schaback, R., 1995, Creating surfaces from scattered data using radial basis functions, 477
Schaback, Robert, 1995, Multivariate interpolation and approximation by translates of a basis function, 491
Schaback, R., 1996, Approximation by radial basis functions with finitely many centers, Constr. Approx., 12, 331, 10.1007/s003659900017
Schaback, Robert, 1996, Operators on radial functions, J. Comput. Appl. Math., 73, 257, 10.1016/0377-0427(96)00047-7
Wendland, Holger, 1995, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math., 4, 389, 10.1007/BF02123482
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.
H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, Journal of Approx. Theory 93 (1998), pp 258-272.