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

Brezzi, Franco; Lipnikov, Konstantin; Shashkov, Mikhail

doi:10.1137/040613950

Cited by 329 publications

(385 citation statements)

References 20 publications

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“…Eymard et al [36], Aavatsmark et al [1], or Droniou and Eymard [32], mimetic finite difference; cf. Brezzi et al [21], covolume; cf. Chou et al [27], and others.…”

Section: Introductionmentioning

confidence: 98%

Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods

Vohralı́k

2010

Math. Comp.

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Abstract. We derive in this paper a unified framework for a priori and a posteriori error analysis of mixed finite element discretizations of second-order elliptic problems. It is based on the classical primal weak formulation, the postprocessing of the potential proposed in [T. Arbogast and Z. Chen, On the implementation of mixed methods as nonconforming methods for secondorder elliptic problems, Math. Comp. 64 (1995), 943-972], and the discrete Friedrichs inequality. Our analysis in particular avoids any explicit use of the uniform discrete inf-sup condition and in a straightforward manner and under minimal necessary assumptions, known convergence and superconvergence results are recovered. The same framework then turns out to lead to optimal a posteriori energy error bounds. In particular, estimators for all families and orders of mixed finite element methods on grids consisting of simplices or rectangular parallelepipeds are derived. They give a guaranteed and fully computable upper bound on the energy error, represent error local lower bounds, and are robust under some conditions on the diffusion-dispersion tensor. They are thus suitable for both overall error control and adaptive mesh refinement. Moreover, the developed abstract framework and a posteriori error estimates are quite general and apply to any locally conservative method. We finally prove that in parallel and simultaneously in converse to Galerkin finite element methods, under some circumstances, the weak solution is the orthogonal projection of the postprocessed mixed finite element approximation onto the H 1 0 (Ω) space and also establish several links between mixed finite element approximations and some generalized weak solutions.

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“…Eymard et al [36], Aavatsmark et al [1], or Droniou and Eymard [32], mimetic finite difference; cf. Brezzi et al [21], covolume; cf. Chou et al [27], and others.…”

Section: Introductionmentioning

confidence: 98%

Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods

Vohralı́k

2010

Math. Comp.

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“…However, the monotonicity of the lumped MHFE method on general quadrangular meshing can be improved by considering each quadrilateral as a macroelement of triangles. The idea of fictitious subdivision into triangles has already been discussed for mixed finite elements [Brezzi and Fortin, 1991;Kuznetsov and Repin, 2003;Kuznetsov and Repin, 2005;Jaffré et al, 2006] and used in the mimetic finite element/finite difference methods [Kuznetsov et al, 2004;Lipnikov et al, 2006;Brezzi et al, 2005aBrezzi et al, , 2005b.…”

Section: A Technique For Improving the Stability Of Quadrangular Mhfementioning

confidence: 99%

“…(4) Anisotropic discontinuous conductivity tensors are treated in a consistent way, like with finite elements. Note that recently, other conservative numerical methods for flow in porous media have been developed [Edwards, 2002], such as the control volume mixed finite element method [Cai et al, 1997;Klausen and Russell, 2005], the support operator method [Hyman et al, 1997;Berndt et al, 2001], the multipoint flux approximation method [Aavatsmark et al, 1994;Edwards andRogers, 1994, 1998;Aavatsmark, 2002], and the mimetic finite difference method [Kuznetsov et al, 2004;Brezzi et al, 2005aBrezzi et al, , 2005bLipnikov et al, 2006].…”

Section: Introductionmentioning

confidence: 99%

Mixed finite elements for solving 2-D diffusion-type equations

2010

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[1] Mixed finite elements are a numerical method becoming more and more popular in geosciences. This method is well suited for solving elliptic and parabolic partial differential equations, which are the mathematical representation of many problems, for instance, flow in porous media, diffusion/ dispersion of solutes, and heat transfer, among others. Mixed finite elements combine the advantages of finite elements by handling complex geometry domains with unstructured meshes and full tensor coefficients and advantages of finite volumes by ensuring mass conservation at the element level. In this work, a physically based presentation of mixed finite elements is given, and the main approximations or reformulations made to improve the efficiency of the method are detailed. These approximations or reformulations exhibit links with other numerical methods (nonconforming finite elements, finite differences, finite volumes, and multipoint flux methods). Some improvements of the mixed finite element method are suggested, especially to avoid oscillations for transient simulations and distorted quadrangular grids.

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“…Further connections between direct and conforming methods can be established by choosing specific quadrature points to compute the integrals in the conforming method [12,13,19]. Note that quadrature selection can be interpreted as yet another choice for the reconstruction operator.…”

Section: Conforming Mimetic Discretizationmentioning

confidence: 99%

“…Further research also revealed connections between some compatible methods. For instance, mimetic FD for the Poisson equation can be obtained from mixed FE by quadrature choice [12,13,19]. Another example is the equivalence between a covolume method and the classical Marker-and-Cell (MAC) scheme on uniform grids [43] and the analysis of [39] that relates finite volume and finite elements by using the concept of a "spread cell".…”

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confidence: 99%

Principles of Mimetic Discretizations of Differential Operators

Bochev

Hyman

The IMA Volumes in Mathematics and Its Applications

158

206

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Abstract. Compatible discretizations transform partial differential equations to discrete algebraic problems that mimic fundamental properties of the continuum equations. We provide a common framework for mimetic discretizations using algebraic topology to guide our analysis. The framework and all attendant discrete structures are put together by using two basic mappings between differential forms and cochains. The key concept of the framework is a natural inner product on cochains which induces a combinatorial Hodge theory on the cochain complex. The framework supports mutually consistent operations of differentiation and integration, has a combinatorial Stokes theorem, and preserves the invariants of the De Rham cohomology groups. This allows, among other things, for an elementary calculation of the kernel of the discrete Laplacian. Our framework provides an abstraction that includes examples of compatible finite element, finite volume, and finite difference methods. We describe how these methods result from a choice of the reconstruction operator and explain when they are equivalent. We demonstrate how to apply the framework for compatible discretization for two scalar versions of the Hodge Laplacian.

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

Abstract: 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.

Cited by 329 publications

References 20 publications

Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods

Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods

Mixed finite elements for solving 2-D diffusion-type equations

Principles of Mimetic Discretizations of Differential Operators

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