Convergence of a Mimetic Finite Difference Method for Static Diffusion Equation

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.

A new mimetic scheme for the acoustic wave equation

“…The general expression (5) is an original contribution of this article, from a notational point of view, that summarizes in a single equation the discrete version of Green-Gauss-Stokes' theorem for any space dimension (n = 1, 2, and 3). In the particular case of n = 1, equation (5) agrees with one dimensional expressions previously reported in the technical literature [1,12,13]. The Green-Gauss-Stokes' theorem (5) provides the following explicit expression for the boundary operator,…”

“…One dimensional second order mimetic discretizations for the gradient and divergence operators have been well documented in [1,12,13]. Their description will be briefly presented, for the sake of completeness, in reference to the one dimensional uniform staggered grid displayed in figure 1.…”

“…Mimetic operators D s nd , G s nd , and B s nd are the only ones required by the discretization of the acoustic wave equation. Their formulation and formal description in a general multidimensional context, represents an important advance in the development of mimetic method as described in [1,12,13].…”

“…Second order mimetic schemes for the acoustic wave equation based on operators described in [1,12,13] have not been reported in the technical literature, in spite of higher order mimetic operators have been used on elastic applications [1,7,9]. The main purpose of this article is to formulate, analyze, and test a new second order mimetic scheme for the scalar acoustic wave equation based on operators given in [1,12,13]. We express the mimetic discretizations of gradient and divergence operators in a tensor form valid for any space dimension, which gives this new method a great generality.…”

“…The first section is this introduction and bibliographical review. Then, the second section presents our discrete tensor formulation of mimetic gradient and divergence operators based on the Kronecker product of the one dimensional operators proposed in [12,13]. Next, these tensor operators are used in the formulation of our new mimetic scheme for the acoustic wave equation in section three.…”