tailieunhanh - Báo cáo hóa học: " Research Article Convergence of a Mimetic Finite Difference Method for Static Diffusion Equation"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Convergence of a Mimetic Finite Difference Method for Static Diffusion Equation | Hindawi Publishing Corporation Advances in Difference Equations Volume 2007 Article ID 12303 12 pages doi 2007 12303 Research Article Convergence of a Mimetic Finite Difference Method for Static Diffusion Equation J. M. Guevara-Jordan S. Rojas M. Freites-Villegas and J. E. Castillo Received 23 January 2007 Revised 2 April 2007 Accepted 19 April 2007 Recommended by Panayiotis D. Siafarikas 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. Copyright 2007 J. M. Guevara-Jordan et al. 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. 1. Introduction Nowadays much effort has been devoted to create a discrete analog of vector and tensor calculus that could be used to accurately approximate continuum models for a wide range of physical and engineering problems which preserves in a discrete sense symmetries and conservation laws that are true in the continuum 1 2 .