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

Victor Bayona Revilla, Natasha Flyer, Bengt Fornberg, Gregory A. Barnett

Research output: Contribution to journalArticlepeer-review

72 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)257-273
Number of pages17
JournalJournal of Computational Physics
Volume332
DOIs
Publication statusPublished - 1 Mar 2017

Keywords

  • Elliptic PDEs
  • Meshless
  • Polyharmonic splines
  • Polynomials
  • RBF-FD
  • Runge's phenomenon

ASJC Scopus subject areas

  • Physics and Astronomy (miscellaneous)
  • Computer Science Applications

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