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Theoretical analysis of numerical integration in Galerkin meshless methods

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Abstract

In this paper, we study effects of numerical integration on Galerkin meshless methods for solving elliptic partial differential equations with Neumann boundary conditions. The shape functions used in the meshless methods reproduce linear polynomials. The numerical integration rules are required to satisfy the so-called zero row sum condition of stiffness matrix, which is also used by Babuška et al. (Int. J. Numer. Methods Eng. 76:1434–1470, 2008). But the analysis presented there relies on a certain property of the approximation space, which is difficult to verify. The analysis in this paper does not require this property. Moreover, the Lagrange multiplier technique was used to handle the pure Neumann condition. We have also identified specific numerical schemes, diagonal elements correction and background mesh integration, that satisfy the zero row sum condition. The numerical experiments are carried out to verify the theoretical results and test the accuracy of the algorithms.

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Authors and Affiliations

  1. Department of Scientific Computing and Computer Applications, Sun Yat-sen University, Guangzhou, 510275, P.R. China

    Qinghui Zhang

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  1. Qinghui Zhang
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Correspondence to Qinghui Zhang.

Additional information

Communicated by Ragnar Winther.

This research was partially supported by the China Scholarship Council through the State Scholarship Program and the Natural Science Foundation of China under grants 11001282 and 11071264.

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Zhang, Q. Theoretical analysis of numerical integration in Galerkin meshless methods. Bit Numer Math 51, 459–480 (2011). https://doi.org/10.1007/s10543-010-0291-3

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  • DOI: https://doi.org/10.1007/s10543-010-0291-3

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