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.
Highlights
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]
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
This endeavor has led to the formulation of a set of mimetic finite difference discretization schemes to find high-order numerical solution of partial differential equations [3, 4]
Summary
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] This endeavor has led to the formulation of a set of mimetic finite difference discretization schemes to find high-order numerical solution of partial differential equations [3, 4]. We will provide a rigorous proof of quadratic convergence for a particular and unique mimetic finite difference method for the steady-state diffusion equation based on the second-order discrete gradient and divergence operators obtained and studied in [5,6,7]. It has been reported that the problem posed by (2.1) and (2.2) has a unique solution when α’s coefficients are not null [12, 13]
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