We propose a multiscale approximation method for constructing numerical solutions to elliptic partial differential equations on a bounded domain. The approximate solution is constructed in a multi-level fashion, with each level using compactly supported radial basis functions on an increasingly fine mesh. Numerical experiments support the theoretical results. References S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics. Springer, 3rd edition, 2008. doi:10.1007/978-1-4757-3658-8. C. S. Chen, M. Ganesh, M. A. Golberg, and A. H.-D. Cheng. Multilevel compact radial functions based computational schemes for some elliptic problems. Computers and Mathematics with Applications, 43:359--378, 2002. doi:10.1016/S0898-1221(01)00292-9. A. Chernih and Q. T. Le Gia. Multiscale methods with compactly supported radial basis functions for Galerkin approximation of elliptic PDEs. submitted, 2012. G. E. Fasshauer. Meshfree Approximation Methods with Matlab, volume 6 of Interdisciplinary Mathematical Sciences. World Scientific Publishing Co., Singapore, 2007. G. E. Fasshauer and J. G. Zhang. On choosing `optimal' shape parameters for RBF approximation. Numerical Algorithms, 45:345--368, 2007. doi:10.1007/s11075-007-9072-8. P. Giesl and H. Wendland. Meshless collocation: Error estimates with application to dynamical systems. SIAM Journal on Numerical Analysis, 45:1723--1741, 2006. doi:10.1137/060658813. Q. T. Le Gia, I. H. Sloan, and H. Wendland. Multiscale RBF collocation for solving PDEs on spheres. Numerische Mathematik, 121:99--125, 2012. doi:10.1007/s00211-011-0428-6. S. Rippa. An algorithm for selecting a good value of the parameter c in radial basis function interpolation. Advance in Computational Mathematics, 11:193--210, 1999. doi:10.1023/A:1018975909870. E. M. Stein. Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, New Jersey, 1970. doi:10.1090/pspum/010/0482394. H. Wendland. Meshless Galerkin methods using radial basis functions. Mathematics of Computation, 68(228):1521--1531, 1998. doi:10.1090/S0025-5718-99-01102-3. H. Wendland. Numerical solution of variational problems by radial basis functions. In Charles K. Chui and Larry L. Schumaker, editors, Approximation Theory IX, Volume 2: Computational Aspects, pages 361--368. Vanderbilt University Press, 1998. H. Wendland. Scattered Data Approximation, volume 17 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, 2005. doi:10.1017/CBO9780511617539.
@article{6304,
title = {Multiscale methods with compactly supported radial basis functions for elliptic partial differential equations on bounded domains},
journal = {ANZIAM Journal},
volume = {53},
year = {2013},
doi = {10.21914/anziamj.v54i0.6304},
language = {EN},
url = {http://dml.mathdoc.fr/item/6304}
}
Chernih, Andrew; Le Gia, Quoc Thong. Multiscale methods with compactly supported radial basis functions for elliptic partial differential equations on bounded domains. ANZIAM Journal, Tome 53 (2013) . doi : 10.21914/anziamj.v54i0.6304. http://gdmltest.u-ga.fr/item/6304/