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Adaptive procedures for meshfree RBF unsymmetric and symmetric collocation methods

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  • Cavoretto, Roberto
  • De Rossi, Alessandra

Abstract

In this article we present an adaptive scheme for solving radial basis function collocation problems, which involve elliptic partial differential equations. The proposed algorithm is applied to two usual numerical methods, known as the nonsymmetric Kansa’s method and the symmetric Hermite-based approach. Basically, the refinement algorithm is firstly characterized by the use of an adaptive superimposition scheme, in which an error estimate compares two approximate solutions computed on a coarser and a finer set of collocation points, and then on a modified adaptive residual subsampling scheme. Blending these computational techniques we detect the areas that need to be refined, also having the chance to further add and/or remove adaptively collocation points. Our study is supported by several numerical results, which illustrate the performance of our iterative algorithm.

Suggested Citation

  • Cavoretto, Roberto & De Rossi, Alessandra, 2020. "Adaptive procedures for meshfree RBF unsymmetric and symmetric collocation methods," Applied Mathematics and Computation, Elsevier, vol. 382(C).
    • Handle: RePEc:eee:apmaco:v:382:y:2020:i:c:s0096300320303180
    DOI: 10.1016/j.amc.2020.125354
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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.
    2. Kaennakham, S. & Chuathong, N., 2019. "An automatic node-adaptive scheme applied with a RBF-collocation meshless method," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 102-125.
    3. 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).
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    Cited by:

    1. Chih-Yu Liu & Cheng-Yu Ku, 2022. "A Simplified Radial Basis Function Method with Exterior Fictitious Sources for Elliptic Boundary Value Problems," Mathematics, MDPI, vol. 10(10), pages 1-23, May.

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