On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs

JOURNAL OF COMPUTATIONAL PHYSICS

卷:332

Article

Bayona, Victor¹  Flyer, Natasha²  Fornberg, Bengt³  Barnett, Gregory A.⁴ 

[1] Univ Bath, Dept Math Sci, Bath BA2 7AY, Avon, England

[2] Natl Ctr Atmospher Res, Inst Math Appl Geosci, Boulder, CO 80305 USA

[3] Univ Colorado, Dept Appl Math, Boulder, CO 80309 USA

[4] Sandia Natl Labs, POB 5800, Albuquerque, NM 87185 USA

关键词: Elliptic PDEs;    RBF-FD;    Polynomials;    Polyharmonic splines;    Runge's phenomenon;    Meshless;   

DOI  :  10.1016/j.jcp.2016.12.008

来源: Elsevier

【 摘 要 】

RBF-generated finite differences (RBF-FD) have in the last decade emerged as a very powerful and flexible numerical approach for solving a wide range of PDEs. We find in the present study that combining polyharmonic splines (PHS) with multivariate polynomials offers an outstanding combination of simplicity, accuracy, and geometric flexibility when solving elliptic equations in irregular (or regular) regions. In particular, the drawbacks on accuracy and stability due to Runge's phenomenon are overcome once the RBF stencils exceed a certain size due to an underlying minimization property. Test problems include the classical 2-D driven cavity, and also a 3-D global electric circuit problem with the earth's irregular topography as its bottom boundary. The results we find are fully consistent with previous results for data interpolation. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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