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Error indicators and refinement strategies for solving Poisson problems through a RBF partition of unity collocation scheme

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  • Cavoretto, Roberto
  • De Rossi, Alessandra

Abstract

In this article adaptive refinement algorithms are presented to solve Poisson problems by a radial basis function partition of unity (RBF-PU) collocation scheme. Since in this context the problem of constructing an adaptive discretization method to be really effective is still open, we propose some error indicators and refinement strategies, so that each of these two essential ingredients takes advantage of the potentiality of the other one. More precisely, the refinement techniques coupled with a local error indicator is an ad-hoc strategy for the RBF-PU method. The resulting scheme turns out to be flexible and the use of efficient searching procedures enables us a fast detection of the regions that adaptively need the addition/removal of points. Several numerical experiments and applications support our study by illustrating the performance of our adaptive algorithms.

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  • Cavoretto, Roberto & De Rossi, Alessandra, 2020. "Error indicators and refinement strategies for solving Poisson problems through a RBF partition of unity collocation scheme," Applied Mathematics and Computation, Elsevier, vol. 369(C).
    • Handle: RePEc:eee:apmaco:v:369:y:2020:i:c:s0096300319308161
    DOI: 10.1016/j.amc.2019.124824
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    References listed on IDEAS

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    1. Oanh, Dang Thi & Davydov, Oleg & Phu, Hoang Xuan, 2017. "Adaptive RBF-FD method for elliptic problems with point singularities in 2D," Applied Mathematics and Computation, Elsevier, vol. 313(C), pages 474-497.
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    Cited by:

    1. Cavoretto, Roberto, 2022. "Adaptive LOOCV-based kernel methods for solving time-dependent BVPs," Applied Mathematics and Computation, Elsevier, vol. 429(C).
    2. Nikan, O. & Avazzadeh, Z., 2022. "A locally stabilized radial basis function partition of unity technique for the sine–Gordon system in nonlinear optics," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 199(C), pages 394-413.
    3. Cavoretto, Roberto & De Rossi, Alessandra, 2020. "Adaptive procedures for meshfree RBF unsymmetric and symmetric collocation methods," Applied Mathematics and Computation, Elsevier, vol. 382(C).
    4. Cavoretto, R. & De Rossi, A. & Perracchione, E., 2023. "Learning with Partition of Unity-based Kriging Estimators," Applied Mathematics and Computation, Elsevier, vol. 448(C).
    5. Nikan, O. & Avazzadeh, Z., 2021. "A localisation technique based on radial basis function partition of unity for solving Sobolev equation arising in fluid dynamics," Applied Mathematics and Computation, Elsevier, vol. 401(C).

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    1. Cavoretto, Roberto & De Rossi, Alessandra, 2020. "Adaptive procedures for meshfree RBF unsymmetric and symmetric collocation methods," Applied Mathematics and Computation, Elsevier, vol. 382(C).

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