Open Access

Convergence of a Mimetic Finite Difference Method for Static Diffusion Equation

  • J. M. Guevara-Jordan1Email author,
  • S. Rojas2,
  • M. Freites-Villegas3 and
  • J. E. Castillo4
Advances in Difference Equations20072007:012303

https://doi.org/10.1155/2007/12303

Received: 23 January 2007

Accepted: 19 April 2007

Published: 7 June 2007

Abstract

The numerical solution of partial differential equations with finite differences mimetic methods that satisfy properties of the continuum differential operators and mimic discrete versions of appropriate integral identities is more likely to produce better approximations. Recently, one of the authors developed a systematic approach to obtain mimetic finite difference discretizations for divergence and gradient operators, which achieves the same order of accuracy on the boundary and inner grid points. This paper uses the second-order version of those operators to develop a new mimetic finite difference method for the steady-state diffusion equation. A complete theoretical and numerical analysis of this new method is presented, including an original and nonstandard proof of the quadratic convergence rate of this new method. The numerical results agree in all cases with our theoretical analysis, providing strong evidence that the new method is a better choice than the standard finite difference method.

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Authors’ Affiliations

(1)
Departamento de Matemáticas, Universidad Central de Venezuela
(2)
Departamento de Física, Universidad Simón Bolívar
(3)
Departamento de Matemática y Física, Universidad Pedagógica Experimental Libertador
(4)
Computational Science Research Center, San Diego State University

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Copyright

© J. M. Guevara-Jordan et al. 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.