A new discretization methodology for diffusion problems on generalized polyhedral meshes

By Franco Brezzi, Konstantin Lipnikov, Mikhail Yu. Shashkov, Valeria Simoncini. 2007

148 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/94af2d0a6635ccecea510b16d5e70f81e80f9142

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In Computational Mechanics

In Computational Theory and Mathematics

In Numerical Analysis

Citations in the corpus

A unified approach for handling convection terms in finite volumes and mimetic discretization methods for elliptic problems (2011, methodology): “In this framework, we mention, for instance, the Discontinuous Galerkin Method, [5, 44] and references therein, the Multi-Point Flux Approximation [1, 2, 48], the Mimetic Finite Difference Method, [8, 11–14,18, 20–23, 37, 40, 41] and references therein, the Hybrid Finite Volume Method [36], and the ”
Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods (2016, methodology): “Recalling the equivalence result proved in [21] for Mixed [20] and Hybrid Finite Volumes and Mimetic Finite Differences [5, 6, 27], the HHO methods with k “ 0 and l “ ́1 or l “ 0 lead in fact to the same family of methods (up to equivalent choices of stabilization).”
Mimetic finite differences for elliptic problems (2009, methodology): “As in [5,7], here we build again a family of discretization methods which is reduced to the standard P1 finite element method [9] in the case of simplicial meshes.”
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].”
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].”
Topology optimization using polytopes (2013, methodology): “The VEM originated from Mimetic Finite Difference (MFD) methods which have been successfully applied to diffusion [26], fluid flow [27] and elasticity problems [28].”
Virtual Elements for Linear Elasticity Problems (2013, background): “The local stiffness matrices M ∈ RNK×NK corresponding to the local bilinear forms ah (·, ·) , K ∈ Th, are implemented following the guidelines in [14, 7].”
An a posteriori error estimator for the mimetic finite difference approximation of elliptic problems (2008, methodology): “As shown in [16], Assumptions (S1)–(S2) and a mild additional positivity condition makes it possible to reformulate the scalar product given by matrix (5) as”
Mimetic finite difference method (2014, methodology): “The subsequent papers [57] [58] develop a new approach to the construction of derived mimetic operators on arbitrary polyhedral meshes.”
MIMETIC FINITE DIFFERENCE APPROXIMATION OF FLOWS IN FRACTURED POROUS MEDIA (2016, methodology): “Thanks to their great flexibility and their capability of preserving the fundamental properties of the underlying physical and mathematical models, the use of MFD methods has remarkably increased in the last years, with application to diffusion-type problems [23, 26, 24, 25, 21, 14, 7], electromagne”
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].”
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”
On the Virtual Element Method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes (2013, unknown): “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]).”
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].”