Advances in Difference Equations
Volume 2007 (2007), Article ID 12303, 12 pages
doi:10.1155/2007/12303
Convergence of a mimetic finite difference method for static diffusion equation
J.M. Guevara-Jordan1
, S. Rojas2
, M. Freites-Villegas3
and J.E. Castillo4
1Departamento de Matemáticas, Universidad Central de Venezuela, Apdo. 6228, Carmelitas 1010, Caracas, Venezuela
2Departamento de Física, Universidad Simón Bolívar, Ofic. 220, Sartenejas, Baruta Coding Postal 1082, Edo. Miranda, Venezuela
3Departamento de Matemática y Física, Universidad Pedagógica Experimental Libertador, Avenida Páez, El Paraiso Coding Postal 1020, Caracas, Venezuela
4Computational Science Research Center, San Diego State University, 5500 Campanile Dr., San Diego 92182-1245, CA, USA
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.