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

By Franco Brezzi, Konstantin Lipnikov, Mikhail Yu. Shashkov. SIAM Journal on Numerical Analysis, 2005

387 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/02d3ecdc836a7e1825ab96aa6839af56d78964e1

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In Computational Mechanics

In Computational Theory and Mathematics

In Numerical Analysis

Citations in the corpus

MIXED FINITE ELEMENT METHODS: IMPLEMENTATION WITH ONE UNKNOWN PER ELEMENT, LOCAL FLUX EXPRESSIONS, POSITIVITY, POLYGONAL MESHES, AND RELATIONS TO OTHER METHODS (2013, neutral): “The last goal of the present paper is to, through the preceding developments, show the closeness of mixed finite element methods to various other discretization schemes, namely mimetic finite difference [43, 20, 40], hybrid finite volume [35], mixed finite volume [31], multi-point fluxapproximation ”
A Mimetic discretization method for linear elasticity (2010, neutral): “Mesh notation and assumptions For exposition’s sake, we find it convenient to adopt the mesh notation and assumptions introduced in [16].”
A FAMILY OF MIMETIC FINITE DIFFERENCE METHODS ON POLYGONAL AND POLYHEDRAL MESHES (2005, methodology): “It was shown in [4] that the convergence analysis of a MFD method is reduced to the existence of a solution of a matrix algebraic equation with constraints.”
Residual a posteriori error estimation for the Virtual Element Method for elliptic problems (2015, methodology): “This approach is successful for the MFD method [20] for the diffusion problem in mixed form.”
A weak Galerkin finite element method for the stokes equations (2013, methodology): “Interested readers are referred to [7] for a similar shape regularity assumption for the mimetic finite difference method.”
Multiscale mixed/mimetic methods on corner-point grids (2008, background): “[12] established that, in addition to symmetry, the following conditions are sufficient to ensure stability and convergence in pressure and velocity of the mimetic FDM for very general polygons:”
Basic principles of Virtual Element Methods (2013, background): “Since their very beginning, Mimetic Finite Differences (see for instance [28, 25, 17, 24, 26]), and in particular their mathematical frameworks and settings, have been evolving from the original Finite Difference/Finite Volumes point of view towards more Finite Element-like presentations and analyse”
Physically based modeling in catchment hydrology at 50: Survey and outlook (2015, methodology): “The most recent developments in this area have led to the construction of the mimetic finite difference method and the closely related virtual element method [Brezzi et al., 2005, 2007; Beir~ao da Veiga et al., 2014a].”
Finite element exterior calculus: From hodge theory to numerical stability (2009, methodology): “Other examples of compatible discretization methods for elliptic PDEs are mimetic finite difference methods [16, 27] including covolume methods [74] and the discrete exterior calculus [38].”
A multigrid algorithm for the $p$-version of the Virtual Element Method (2017, methodology): “Examples of methods that have been proposed so far include mimetic finite differences [20, 35, 51, 52], polygonal finite element methods [58, 59], discontinuous-Galerkin finite element methods [4, 5, 9, 16, 37, 38, 40] hybridizable and hybrid high hrder methods [42, 43, 45], BEM-based FEM [56] and g”
Virtual Elements for Linear Elasticity Problems (2013, background): “(For a partial list of citations we refer to [29, 25, 17, 24, 26, 13, 15, 8, 27, 1, 14, 6, 18, 21, 32, 11, 2, 10, 16].”
A Two-Phase Flow Simulation of Discrete-Fractured Media using Mimetic Finite Difference Method (2014, methodology): “[40,41] developed the mimetic finite difference (MFD) method whose formulation is similar to the mixed-hybrid formulation.”
A unified approach for handling convection terms in finite volumes and mimetic discretization methods for elliptic problems (2011, methodology): “Moreover, when the discretization of the convection term increases significantly the numerical diffusion, as for example in the case of the upwind scheme, the hybrid version of the HMM method is likely to be less diffusive than that provided by (2.39)-(2.40).”
Mixed finite elements for solving 2‐D diffusion‐type equations (2010, methodology): “…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., 2005a, 2005b].”