A Multiscale RBF Collocation Method for the Numerical Solution of Partial Differential Equations

Zhiyong Liu and Qiuyan Xu
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Zhiyong Liu: School of Mathematical Sciences, Fudan University, Shanghai 200433, China
Qiuyan Xu: school of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China

Mathematics, 2019, vol. 7, issue 10, 1-15

Abstract: In this paper, we derive and discuss the hierarchical radial basis functions method for the approximation to Sobolev functions and the collocation to well-posed linear partial differential equations. Similar to multilevel splitting of finite element spaces, the hierarchical radial basis functions are constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. Compared with the compactly supported radial basis functions approximation and stationary multilevel approximation, the new method can not only solve the present problem on a single level with higher accuracy and lower computational cost, but also produce a highly sparse discrete algebraic system. These observations are obtained by taking the direct approach of numerical experimentation.

Keywords: Kansa method; Sobolev spaces; native spaces; collocation method; radial basis functions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
References: View complete reference list from CitEc
Citations: View citations in EconPapers (4)

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