Abstract
In this paper, we explore the effect of numerical integration on the Galerkin meshless method used to approximate the solution of an elliptic partial differential equation with non-constant coefficients with Neumann boundary conditions. We considered Galerkin meshless methods with shape functions that reproduce polynomials of degree k ≥ 1. We have obtained an estimate for the energy norm of the error in the approximate solution under the presence of numerical integration. This result has been established under the assumption that the numerical integration rule satisfies a certain discrete Green’s formula, which is not problem dependent, i.e., does not depend on the non-constant coefficients of the problem. We have also derived numerical integration rules satisfying the discrete Green’s formula.
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Communicated by Z. Chen.
This research was partially supported by the NSF Grant # DMS-0610778, Natural Science Foundation of China grant 11001282, Fundamental Research Funds for Central Universities and by Prof. Y. Xu’s grant from 985 project at Sun Yat-sen University, Guangdong Provincial Natural Science Foundation of China under grants S2011040003030.
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Zhang, Q., Banerjee, U. Numerical integration in Galerkin meshless methods, applied to elliptic Neumann problem with non-constant coefficients. Adv Comput Math 37, 453–492 (2012). https://doi.org/10.1007/s10444-011-9216-1
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DOI: https://doi.org/10.1007/s10444-011-9216-1
Keywords
- PDE with non-constant coefficients
- Galerkin methods
- Meshless methods
- Quadrature
- Numerical integration
- Error estimates