Multiscale methods with compactly supported radial basis functions for elliptic partial differential equations on bounded domains

Andrew Chernih, Quoc Thong Le Gia


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.

  • 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.


multiscale, collocation, radial basis function, elliptic

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