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RBF-Based Partition of Unity Methods for Elliptic PDEs: Adaptivity and Stability Issues Via Variably Scaled Kernels

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Abstract

We investigate adaptivity issues for the approximation of Poisson equations via radial basis function-based partition of unity collocation. The adaptive residual subsampling approach is performed with quasi-uniform node sequences leading to a flexible tool which however might suffer from numerical instability due to ill-conditioning of the collocation matrices. We thus develop a hybrid method which makes use of the so-called variably scaled kernels. The proposed algorithm numerically ensures the convergence of the adaptive procedure.

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Acknowledgements

We sincerely thank the reviewers for their insightful comments. This research has been accomplished within Rete ITaliana di Approssimazione (RITA) and supported by GNCS-IN\(\delta \)AM. The first author was partially supported by the research project Approximation by radial basis functions and polynomials: applications to CT, MPI and PDEs on manifolds, No. DOR1695473. The third author was partially supported by the research project Radial basis functions approximations: stability issues and applications, No. BIRD167404.

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Authors and Affiliations

  1. Dipartimento di Matematica, “Tullio Levi-Civita”, Università di Padova, Padua, Italy

    S. De Marchi, A. Martínez & E. Perracchione

  2. Dipartimento di Matematica e Applicazioni, Università di Milano - Bicocca, Milan, Italy

    M. Rossini

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  1. S. De Marchi
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  2. A. Martínez
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  4. M. Rossini
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Correspondence to E. Perracchione.

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De Marchi, S., Martínez, A., Perracchione, E. et al. RBF-Based Partition of Unity Methods for Elliptic PDEs: Adaptivity and Stability Issues Via Variably Scaled Kernels. J Sci Comput 79, 321–344 (2019). https://doi.org/10.1007/s10915-018-0851-2

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