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

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• 作者：Victor Bayona^a ; ^{victor.bayona.revilla@gmail.com} ; Natasha Flyer^b ; ^{flyer@ucar.edu} ; Bengt Fornberg^c ; ^{fornberg@colorado.edu} ; Gregory A. Barnett^d ; ^{gabarnettjr@gmail.com}
• 关键词：Elliptic PDEs ; RBF-FD ; Polynomials ; Polyharmonic splines ; Runge's phenomenon ; Meshless
• 刊名：Journal of Computational Physics
• 出版年：2017
• 出版时间：1 March 2017
• 年：2017
• 卷：332
• 期：Complete
• 页码：257-273
• 全文大小：3473 K
• 卷排序：332

文摘

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.