Computing singular solutions to partial differential equations by Taylor series

Jie Yang; Heng Hu; Michel Potier-Ferry

doi:10.1016/j.crme.2018.04.003

Model reduction, data-based and advanced discretization in computational mechanics

Computing singular solutions to partial differential equations by Taylor series

Jie Yang ^1,²; Heng Hu ¹ ; Michel Potier-Ferry ²

¹ School of Civil Engineering, Wuhan University, 8 South Road of East Lake, Wuchang, 430072 Wuhan, PR China
² Laboratoire d'étude des microstructures et de mécanique des matériaux, LEM3, UMR CNRS 7239, Université de Lorraine, île du Saulcy, 57045 Metz cedex 01, France

Comptes Rendus. Mécanique, Volume 346 (2018) no. 7, pp. 603-614.

Abstract

The Taylor Meshless Method (TMM) is a true meshless integration-free numerical method for solving elliptic Partial Differential Equations (PDEs). The basic idea of this method is to use high-order polynomial shape functions that are approximated solutions to the PDE and are computed by the technique of Taylor series. Currently, this new method has proved robust and efficient, and it has the property of exponential convergence with the degree, when solving problems with smooth solutions. This exponential convergence is no longer obtained for problems involving cracks, corners or notches. On the basis of numerical tests, this paper establishes that the presence of a singularity leads to a worsened convergence of the Taylor series, but highly accurate solutions can be recovered by including a few singular solutions in the basis of shape functions.

Received: 2018-02-03
Accepted: 2018-03-09
Published online: 2018-04-27

DOI: 10.1016/j.crme.2018.04.003

Keywords: Taylor series, Meshless, Singular shape functions, Angular domain

Author's affiliations:

Jie Yang ^{1, 2}; Heng Hu ¹; Michel Potier-Ferry ²

¹ School of Civil Engineering, Wuhan University, 8 South Road of East Lake, Wuchang, 430072 Wuhan, PR China
² Laboratoire d'étude des microstructures et de mécanique des matériaux, LEM3, UMR CNRS 7239, Université de Lorraine, île du Saulcy, 57045 Metz cedex 01, France

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     title = {Computing singular solutions to partial differential equations by {Taylor} series},
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     language = {en},
}

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Jie Yang; Heng Hu; Michel Potier-Ferry. Computing singular solutions to partial differential equations by Taylor series. Comptes Rendus. Mécanique, Volume 346 (2018) no. 7, pp. 603-614. doi : 10.1016/j.crme.2018.04.003. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2018.04.003/

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