A FAMILY OF MIMETIC FINITE DIFFERENCE METHODS ON POLYGONAL AND POLYHEDRAL MESHES

By Franco Brezzi, Konstantin Lipnikov, Valeria Simoncini. 2005

421 citations recorded by the source corpus.

Subfields: Computational Mechanics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/d0da3bce0c25e522fa7444a9339888301094e5d0

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In Computational Mechanics

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Citations in the corpus

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.”
Mimetic finite differences for elliptic problems (2009, methodology): “Then, the general form of the matrix MP is (see [7] for more details) MP = 1 |P | A1 K̃ −1 A T 1 + D1Ũ D T 1 , (7.”
New perspectives on polygonal and polyhedral finite element methods (2014, methodology): “Then, this method evolved into the Mimetic Finite Difference method, which first extended the discretization of differential operators to unstructured meshes with elements of very general shape [98, 99, 100, 101, 102, 103].”
On the Virtual Element Method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes (2013, positive): “Mimetic ﬁnite diﬀerence (MFD) methods, capable of handling general three-dimensional meshes, are also the subject of active research and have been successfully applied to diﬀusion, elasticity, and ﬂuid ﬂow problems (see, for example, [16, 17, 18, 8, 5]).”
A Mimetic discretization method for linear elasticity (2010, methodology): “Following the same proof as in [8,17], we have N R = |E|K, (4.”
Monotone nonlinear finite‐volume method for nonisothermal two‐phase two‐component flow in porous media (2017, methodology): “Due to the additional requirement of mass-conservation, the most commonly used methods for solving flow problems in porous media are multi-point flux approximation (MPFA) methods [1–5], mixed finite element (MFE) methods [6, 7], or mimetic finite difference (MFD) methods [8, 9].”
Virtual Element Methods for general second order elliptic problems on polygonal meshes (2014, neutral): “See in particular, for Mimetic Finite Diﬀerences, [3], [13], [14], [17], [24], [26], [27], [25], and mostly [16], [49] (and the references therein), and [1], [2], [19], [10], [11], [18], [28], [29], [45], [46], [50], [53], [64] for Virtual Elements.”
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].”
Equivalent projectors for virtual element methods (2013, background): “The origin of Virtual Elements, historically, is clearly from the classical Mimetic Finite Difference Method (see for instance [30, 26, 18, 27, 28]), and from their subsequent mathematical frameworks and settings [14, 16, 9, 15].”
Multiscale mixed/mimetic methods on corner-point grids (2008, methodology): “Implementation of Mm Explicit formulae for computing Mm-matrices that obey (12)–(13) in the case where the variables vm i represent net face velocities were given in [15].”
Basic principles of Virtual Element Methods (2013, background): “Note however that, in particular, here the construction does not require to choose a set of linearly independent vectors among the ones that are locally orthogonal to polynomials, as done, from the very beginning of these constructions, in [15].”
The Hitchhiker's Guide to the Virtual Element Method (2014, background): “On the other hand, many of its aspects are closely connected with Finite Volume Methods (see for instance [19,20,28,32]) and in particular with Mimetic Finite Differences (MFD) [2–4, 7, 10, 12, 14–16, 18, 22, 23, 25–27].”