Extending the local radial basis function collocation methods for solving semi-linear partial differential equations

G. Gutierrez, O. R. Baquero, J. M. Valencia, W. F. Florez

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    2 Scopus citations

    Abstract

    This work addresses local radial basis function (RBF) collocation methods for solving a major class of non-linear boundary value problems, i.e., Lu = f(x, u) being f a non-linear function of u. This class of problems has been largely analyzed in the BEM community. To our knowledge, few works are reported where the local RBF collocation methods (LRBFCM) based on the generalized Hermite RBF interpolation (double collocation) have been extended successfully to solve semi-linear problems even when extending to more complex nonlinear cases are not reported yet. The studied schemes are based on a strong-form approach of the PDE and an overlapping multi-domain procedure combining with standard iterative schemes. At each sub-domain, a locally meshless approximation solution by a standard or Hermite RBF expansion can be constructed.We studied also the performance respect to the shape parameter of RBF. It is confirmed that the local RBF double collocation can improve greatly the accuracy order. Some 2D benchmark problems with mixed boundary conditions showing the accuracy, convergence property and implementation issues of LRBFCM are presented.

    Original languageEnglish
    Title of host publicationMesh Reduction Methods
    Subtitle of host publicationBEM/MRM XXXI
    Pages117-128
    Number of pages12
    DOIs
    StatePublished - 2009
    Event31st International Conference on Boundary Elements and Other Mesh Reduction Methods, BEM/MRM 31 - New Forest, United Kingdom
    Duration: 2 Sep 20094 Sep 2009

    Publication series

    NameWIT Transactions on Modelling and Simulation
    Volume49
    ISSN (Print)1743-355X

    Conference

    Conference31st International Conference on Boundary Elements and Other Mesh Reduction Methods, BEM/MRM 31
    Country/TerritoryUnited Kingdom
    CityNew Forest
    Period2/09/094/09/09

    Keywords

    • Domain decomposition methods
    • Double collocation
    • Fully Newton method
    • Picard iteration
    • RBF interpolation
    • Semi-linear equation

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