Journal of Scientific Computing

, Volume 65, Issue 3, pp 1240–1269 | Cite as

A Kansa-Radial Basis Function Method for Elliptic Boundary Value Problems in Annular Domains

  • Xiao-Yan Liu
  • Andreas Karageorghis
  • C. S. Chen


We employ a Kansa-radial basis function (RBF) method for the numerical solution of elliptic boundary value problems in annular domains. This discretization leads, with an appropriate selection of collocation points and for any choice of RBF, to linear systems in which the matrices possess block circulant structures. These linear systems can be solved efficiently using matrix decomposition algorithms and fast Fourier transforms. A suitable value for the shape parameter in the various RBFs used is found using the leave-one-out cross validation algorithm. In particular, we consider problems governed by the Poisson equation, the inhomogeneous biharmonic equation and the inhomogeneous Cauchy–Navier equations of elasticity. In addition to its simplicity, the proposed method can both achieve high accuracy and solve large-scale problems. The feasibility of the proposed techniques is illustrated by several numerical examples.


Radial basis functions Poisson equation Biharmonic equation  Cauchy–Navier equations of elasticity Fast Fourier transforms Kansa method 

Mathematics Subject Classification

Primary 65N35 Secondary 65N21 65N38 



The authors wish to thank the referees for their helpful comments and suggestions which resulted in an improved manuscript.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Xiao-Yan Liu
    • 1
  • Andreas Karageorghis
    • 2
  • C. S. Chen
    • 1
    • 3
  1. 1.School of MathematicsTaiyuan University of TechnologyTaiyuanChina
  2. 2.Department of Mathematics and StatisticsUniversity of CyprusNicosiaCyprus
  3. 3.Department of MathematicsUniversity of Southern MississippiHattiesburgUSA

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