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Numerical Computation of Discrete Differential Operators on Non-Uniform Grids

N. Sukumar1, J. E. Bolander1
Department of Civil and Environmental Engineering, One Shields Avenue, University of California, Davis, CA 95616, U.S.A.

Computer Modeling in Engineering & Sciences 2003, 4(6), 691-706. https://doi.org/10.3970/cmes.2003.004.691

Abstract

In this paper, we explore the numerical approximation of discrete differential operators on non-uniform grids. The Voronoi cell and the notion of natural neighbors are used to approximate the Laplacian and the gradient operator on irregular grids. The underlying weight measure used in the numerical computations is the {\em Laplace weight function}, which has been previously adopted in meshless Galerkin methods. We develop a difference approximation for the diffusion operator on irregular grids, and present numerical solutions for the Poisson equation. On regular grids, the discrete Laplacian is shown to reduce to the classical finite difference scheme. Two techniques to compute the nodal (gradient) flux are presented, and benchmark computations in 2-d are performed to demonstrate the accuracy of the schemes. The numerical approximations developed herein are of relevance in the solution of partial differential equations, in methods where local (Laplacian) smoothing is desired, and for applications such as mesh adaptivity in which a posteriori error estimates using stress-based recovery schemes are used.

Keywords

meshless methods, natural neighbor, Voronoi diagram, Laplace interpolant, finite volume, irregular lattice, diffusion, supraconvergence

Cite This Article

Sukumar, N., Bolander, J. E. (2003). Numerical Computation of Discrete Differential Operators on Non-Uniform Grids. CMES-Computer Modeling in Engineering & Sciences, 4(6), 691–706.



This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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