Theoretical analysis of numerical integration in聽Galerkin meshless methods

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• 作者：Qinghui Zhang (1) zhangqh6@mail.sysu.edu.cn
• 关键词：Galerkin method &#8211 ; Meshless methods &#8211 ; Numerical integration &#8211 ; Zero row sum &#8211 ; Convergence order &#8211 ; Background mesh integration
• 刊名：BIT Numerical Mathematics
• 出版时间：June 2011
• 出版年：2011
• 期刊代码：30_0006-3835
• 类别：mat
• 卷：51
• 期：2
• 页码：459-480
• 数据来源：sp

摘要

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.