- Research Article
- Open Access
Convergence of a Mimetic Finite Difference Method for Static Diffusion Equation
Advances in Difference Equations volume 2007, Article number: 012303 (2007)
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.
Morton KW, Mayers DF: Numerical Solution of Partial Differential Equations. CRC Press, Cambridge, UK; 1994:x+227.
Shashkov M: Conservative Finite-Difference Methods on General Grids, Symbolic and Numeric Computation Series. CRC Press, Boca Raton, Fla, USA; 2001.
Castillo JE, Hyman JM, Shashkov M, Steinberg S: Fourth- and sixth-order conservative finite difference approximations of the divergence and gradient. Applied Numerical Mathematics 2001,37(1–2):171–187. 10.1016/S0168-9274(00)00033-7
Lipnikov K, Morel J, Shashkov M: Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes. Journal of Computational Physics 2004,199(2):589–597. 10.1016/j.jcp.2004.02.016
Castillo JE, Grone RD: A matrix analysis approach to higher-order approximations for divergence and gradients satisfying a global conservation law. SIAM Journal on Matrix Analysis and Applications 2003,25(1):128–142. 10.1137/S0895479801398025
Castillo JE, Yasuda M: A comparison of two matrix operator formulations for mimetic divergence and gradient discretizations. Proceedings of the International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA '03), June 2003, Las Vegas, Nev, USA 3: 1281–1285.
Castillo JE, Yasuda M: Linear systems arising for second-order mimetic divergence and gradient discretizations. Journal of Mathematical Modelling and Algorithms 2005,4(1):67–82. 10.1007/s10852-004-3523-1
Freites-Villegas M, Guevara-Jordan JM, Rojas OR, Castillo JE, Rojas S: A mimetic finite difference scheme for solving the steady state diffusion equation with singular sources. In Simulación Numérica y Modelado Computacional. Proceedings of the 7th International Congress of Numerical Methods in Engineering and Applied Science, 2004, San Cristóbal, Venezuela Edited by: Rojo J, Torres MJ, Cerrolaza M. 25–32.
Carslaw HS, Jaeger JC: Conduction of Heat in Solids. 2nd edition. Clarendon Press, Oxford, UK; 1959.
Rojas S, Koplik J: Nonlinear flow in porous media. Physical Review E 1998,58(4):4776–4782. 10.1103/PhysRevE.58.4776
Wilmott P, Howison S, Dewynne J: The Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press, Cambridg, UK; 1995:xiv+317.
Powers DL: Boundary Value Problems. 3rd edition. John Wiley & Sons, New York, NY, USA; 1987.
Thomas JW: Numerical Partial Differential Equations: Finite Difference Methods, Texts in Applied Mathematics. Volume 22. 3rd edition. Springer, New York, NY, USA; 1995:xx+437.
Guevara-Jordan JM, Rojas S, Freites-Villegas M, Castillo JE: A new second order finite difference conservative scheme. Divulgaciones Matemáticas 2005,13(2):107–122.
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Guevara-Jordan, J.M., Rojas, S., Freites-Villegas, M. et al. Convergence of a Mimetic Finite Difference Method for Static Diffusion Equation. Adv Differ Equ 2007, 012303 (2007). https://doi.org/10.1155/2007/12303
- Diffusion Equation
- Discrete Version
- Finite Difference Method
- Static Diffusion
- Integral Identity