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Conformal Mapping for the Efficient Solution of Poisson Problems with the Kansa-RBF Method

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Abstract

We consider the solution of Poisson Dirichlet problems in simply-connected irregular domains. These domains are conformally mapped onto the unit disk and the resulting Poisson Dirichlet problems are solved efficiently using a Kansa-radial basis function (RBF) method with a matrix decomposition algorithm (MDA). In a similar way, we treat Poisson Dirichlet and Poisson Dirichlet–Neumann problems in doubly-connected domains. These domains are mapped onto annular domains by a conformal mapping and the resulting Poisson Dirichlet and Poisson Dirichlet–Neumann problems are solved efficiently using a Kansa-RBF MDA. Several examples demonstrating the applicability of the proposed technique are presented.

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Acknowledgements

The authors are grateful to Professor Nick Papamichael for many enlightening discussions.

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Authors and Affiliations

  1. School of Mathematics, Taiyuan University of Technology, Taiyuan, China

    Xiao-Yan Liu & C. S. Chen

  2. Department of Mathematics, University of Southern Mississippi, Hattiesburg, MS, 39406, USA

    C. S. Chen

  3. Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678, Nicosia, Cyprus

    Andreas Karageorghis

Authors
  1. Xiao-Yan Liu
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  2. C. S. Chen
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  3. Andreas Karageorghis
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Correspondence to Xiao-Yan Liu.

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Liu, XY., Chen, C.S. & Karageorghis, A. Conformal Mapping for the Efficient Solution of Poisson Problems with the Kansa-RBF Method. J Sci Comput 71, 1035–1061 (2017). https://doi.org/10.1007/s10915-016-0330-6

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  • DOI: https://doi.org/10.1007/s10915-016-0330-6

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