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

by Zhiyong Liu 1,2,* and Qiuyan Xu 2
1
School of Mathematical Sciences, Fudan University, Shanghai 200433, China
2
school of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(10), 964; https://doi.org/10.3390/math7100964
Received: 10 September 2019 / Revised: 7 October 2019 / Accepted: 10 October 2019 / Published: 13 October 2019
(This article belongs to the Section Mathematics and Computer Science)
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. View Full-Text
Keywords: Kansa method; Sobolev spaces; native spaces; collocation method; radial basis functions Kansa method; Sobolev spaces; native spaces; collocation method; radial basis functions
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Liu, Z.; Xu, Q. A Multiscale RBF Collocation Method for the Numerical Solution of Partial Differential Equations. Mathematics 2019, 7, 964.

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