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

  • S. De Marchi
  • A. Martínez
  • E. Perracchione
  • M. Rossini
Article
  • 16 Downloads

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.

Keywords

Partition of unity method Radial basis functions Meshfree approximation Elliptic PDEs Variably scaled kernels 

Mathematics Subject Classification

65D05 65D15 65N99 

Notes

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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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • S. De Marchi
    • 1
  • A. Martínez
    • 1
  • E. Perracchione
    • 1
  • M. Rossini
    • 2
  1. 1.Dipartimento di Matematica, “Tullio Levi-Civita”Università di PadovaPaduaItaly
  2. 2.Dipartimento di Matematica e ApplicazioniUniversità di Milano - BicoccaMilanItaly

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