Mimetic Finite Difference Method Research Articles

Mimetic finite difference method for the Stokes problem on polygonal meshes

Various approaches to extend finite element methods to non-traditional elements (general polygons, pyramids, polyhedra, etc.) have been developed over the last decade. The construction of basis functions for such elements is a challenging task and may require extensive geometrical analysis. The mimetic finite difference (MFD) method works on general polygonal meshes and has many similarities with low-order finite element methods. Both schemes try to preserve the fundamental properties of the underlying physical and mathematical models. The essential difference between the two schemes is that the MFD method uses only the surface representation of discrete unknowns to build the stiffness and mass matrices. Since no extension of basis functions inside the mesh elements is required, practical implementation of the MFD method is simple for polygonal meshes that may include degenerate and non-convex elements. In this article, we present a new MFD method for the Stokes problem on arbitrary polygonal meshes and analyze its stability. The method is developed for the general case of tensor coefficients, which allows us to apply it to a linear elasticity problem, as well. Numerical experiments show, for the velocity variable, second-order convergence in a discrete L 2 norm and first-order convergence in a discrete H 1 norm. For the pressure variable, first-order convergence is shown in the L 2 norm.

Convergence analysis of the high-order mimetic finite difference method

We prove second-order convergence of the conservative variable and its flux in the high-order MFD method. The convergence results are proved for unstructured polyhedral meshes and full tensor diffusion coefficients. For the case of non-constant coefficients, we also develop a new family of high-order MFD methods. Theoretical result are confirmed through numerical experiments.

Convergence Analysis of the Mimetic Finite Difference Method for Elliptic Problems

We propose a family of mimetic discretization schemes for elliptic problems including convection and reaction terms. Our approach is an extension of the mimetic methodology for purely diffusive problems on unstructured polygonal and polyhedral meshes. The a priori error analysis relies on the connection between the mimetic formulation and the lowest order Raviart-Thomas mixed finite element method. The theoretical results are confirmed by numerical experiments.

Mimetic finite differences for elliptic problems

We developed a mimetic finite difference method for solving elliptic equations with tensor coefficients on polyhedral meshes. The first-order convergence estimates in a mesh-dependent H1 norm are derived.

Local flux mimetic finite difference methods

We develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes. To approximate the velocity (vector variable), the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n-gonal mesh face in three dimensions. To approximate the pressure (scalar variable), the method uses one degree of freedom per element. A specially chosen quadrature rule for the L 2-product of vector-functions allows for a local flux elimination and reduction of the method to a cell-centered finite difference scheme for the pressure unknowns. Under certain assumptions, first-order convergence is proved for both variables and second-order convergence is proved for the pressure. The assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method. For general polyhedral meshes, non-symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions. Numerical results confirm the theory.

Interpolation-free monotone finite volume method for diffusion equations on polygonal meshes

We developed a new monotone finite volume method for diffusion equations. The second-order linear methods, such as the multipoint flux approximation, mixed finite element and mimetic finite difference methods, are not monotone on strongly anisotropic meshes or for diffusion problems with strongly anisotropic coefficients. The finite volume (FV) method with linear two-point flux approximation is monotone but not even first-order accurate in these cases. The developed monotone method is based on a nonlinear two-point flux approximation. It does not require any interpolation scheme and thus differs from other nonlinear finite volume methods based on a two-point flux approximation. The second-order convergence rate is verified with numerical experiments.

High-order mimetic finite difference method for diffusion problems on polygonal meshes

The mimetic finite difference (MFD) methods mimic important properties of physical and mathematical models. As a result, conservation laws, solution symmetries, and the fundamental identities of the vector and tensor calculus are held for discrete models. The MFD methods retain these attractive properties for full tensor coefficients and arbitrary polygonal meshes which may include non-convex and degenerate elements. The existing MFD methods for solving diffusion-type problems are second-order accurate for the conservative variable (temperature, pressure, energy, etc.) and only first-order accurate for its flux. We developed new high-order MFD methods which are second-order accurate for both scalar and vector variables. The second-order convergence rates are demonstrated with a few numerical examples on randomly perturbed quadrilateral and polygonal meshes.

A Higher-Order Formulation of the Mimetic Finite Difference Method

A new mimetic finite difference method for the diffusion problem is developed by using a linear interpolation for the numerical fluxes. This approach provides a higher-order accurate approximation to the flux of the exact solution. In analogy with the original formulation, a family of local scalar products is constructed to satisfy the fundamental properties of local consistency and spectral stability. The scalar solution field is approximated by a piecewise constant function. A computationally efficient postprocessing technique is also proposed to get a piecewise quadratic polynomial approximation to the exact scalar variable. Finally, optimal convergence rates and accuracy improvement with respect to the lower-order formulation are shown by numerical examples.

A residual based error estimator for the Mimetic Finite Difference method

We present a local error indicator for the Mimetic Finite Difference method for diffusion-type problems on polyhedral meshes. Under essentially the same general hypotheses used in (SIAM J. Numer. Anal. 43:1872–1896, 2005) to show the convergence of the method, we prove the global reliability and local efficiency of the proposed estimator.

Flux reconstruction and solution post-processing in mimetic finite difference methods

We present a post-processing technique for the mimetic finite difference solution of diffusion problems in mixed form. Our post-processing method yields a piecewise linear approximation of the scalar variable that is second-order accurate in the L 2-norm on quite general polyhedral meshes, including non-convex and non-matching elements. The post-processing is based on the reconstruction of vector fields projected onto the mimetic space of vector variables. This technique is exact on constant vector fields and is shown to be independent of the mimetic scalar product choice if a local consistency condition is satisfied. The post-processing method is computationally inexpensive. Optimal performance is confirmed by numerical experiments.

Discrete Orthogonal Decomposition and Variational Fluid Flow Estimation

We exploit the mimetic finite difference method introduced by Hyman and Shashkov to present a framework for estimating vector fields and related scalar fields (divergence, curl) of physical interest from image sequences. Our approach provides a basis for consistent definitions of higher-order differential operators, for the analysis and a novel stability result concerning second-order div-curl regularizers, for novel variational schemes to the estimation of solenoidal (divergence-free) image flows, and to convergent numerical methods in terms of subspace corrections.

Convergence of a Mimetic Finite Difference Method for Static Diffusion Equation

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 mortar mimetic finite difference method on non-matching grids

We consider mimetic finite difference approximations to second order elliptic problems on non-matching multiblock grids. Mortar finite elements are employed on the non-matching interfaces to impose weak flux continuity. Optimal convergence and, in certain cases, superconvergence is established for both the scalar variable and its flux. The theory is confirmed by computational results.

Convergence of the Mimetic Finite Difference Method for Diffusion Problems on Polyhedral Meshes

The stability and convergence properties of the mimetic finite difference method for diffusion-type problems on polyhedral meshes are analyzed. The optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved.

The mimetic finite difference method on polygonal meshes for diffusion-type problems

New mimetic discretizations of diffusion-type equations (for instance, equations modeling single phase Darcy flow in porous media) on unstructured polygonal meshes are derived. The first order convergence rate for the fluid velocity and the second-order convergence rate for the pressure on polygonal, locally refined and non-matching meshes are demonstrated with numerical experiments.

Analysis and optimization of inner products for mimetic finite difference methods on a triangular grid

The support operator method designs mimetic finite difference schemes by first constructing a discrete divergence operator based on the divergence theorem, and then defining the discrete gradient operator as the adjoint operator of the divergence based on the Gauss theorem connecting the divergence and gradient operators, which remains valid also in the discrete case. When evaluating the discrete gradient operator, one needs to define discrete inner products of two discrete vector fields. The local discrete inner product on a given triangle is defined by a 3×3 symmetric positive definite matrix M defined by its six independent elements–parameters. Using the Gauss theorem over our triangle, we evaluate the discrete gradient in the triangle. We require the discrete gradient to be exact for linear functions, which gives us a system of linear equations for elements of the matrix M. This system, together with inequalities which guarantee positive definiteness of the matrix M, results in a one parameter family of inner products which give exact gradients for linear functions. The traditional inner product is a member of this family. The positive free parameter can be used to improve another property of the discrete method. We show that accuracy of the method for quadratic functions improves with decreasing this parameter, however, at the same time, the condition number of the matrix M, which is the local matrix of the linear system for computing the discrete gradient, increases to infinity when the parameter goes to zero, so one needs to choose a compromise between accuracy and solvability of the local system. Our analysis has been performed by computer algebra tools which proved to be essential.

Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes

Mimetic discretizations based on the support-operators methodology are derived for non-orthogonal locally refined quadrilateral meshes. The second-order convergence rate on non-smooth meshes is verified with numerical examples.

Mimetic Finite Difference Methods for Diffusion Equations

This paper reviews and extends the theory and application of mimetic finite difference methods for the solution of diffusion problems in strongly heterogeneous anisotropic materials. These difference operators satisfy the fundamental identities, conservation laws and theorems of vector and tensor calculus on nonorthogonal, nonsmooth, structured and unstructured computational grids. We provide explicit approximations for equations in two dimensions with discontinuous anisotropic diffusion tensors. We mention the similarities and differences between the new methods and mixed finite element or hybrid mixed finite element methods.

The effect of inner products for discrete vector fields on the accuracy of mimetic finite difference methods

The support operators method of discretizing partial differential equations produces discrete analogs of continuum initial boundary value problems that exactly satisfy discrete conservation laws analogous to those satisfied by the continuum system. Thus, the stability of the method is assured, but currently there is no theory that predicts the accuracy of the method on nonuniform grids. In this paper, we numerically investigate how the accuracy, particularly the accuracy of the fluxes, depends on the definition of the inner product for discrete vector fields. We introduce two different discrete inner products, the standard inner product that we have used previously and a new more accurate inner product. The definitions of these inner products are based on interpolation of the fluxes of vector fields. The derivation of the new inner product is closely related to the use of the Piola transform in mixed finite elements. Computing the formulas for the new accurate inner product requires a nontrivial use of computer algebra. From the results of our numerical experiments, we can conclude that using more accurate inner product produces a method with the same order of convergence as the standard inner product, but the constant in error estimate is about three times less. However, the method based on the standard inner product is easier to compute with and less sensitive to grid irregularities, so we recommend its use for rough grids.

Mimetic Finite Difference Methods for Maxwell's Equations and the Equations of Magnetic Diffusion - Abstract

We have constructed mimetic finite difference methods for both the TE and TM modes for 2-D Maxwell's curl equations and equations of magnetic diffusion with discontinuous coefficients on nonorthogonal, nonsmooth grids. The discrete operators were derived using the discrete vector and tensor analysis to satisfy discrete analogs of the main theorems of vector analysis. Because the finite difference methods satisfy these theorems, they do not have spurious solutions and the "divergence-free" conditions for Maxwell's equations are automatically satisfied. The tangential components of the electric field and the normal components of magnetic flux used in the FDM are continuous even across discontinuities. This choice guarantees that problems with strongly discontinuous coefficients are treated properly. Furthermore on rectangular grids the method reduces to the analytically correct averaging for discontinuous coefficients. We verify that the convergence rate was between first and second order on the arbitrary quadrilateral grids and demonstrate robustness of the method in numerical examples.