Equivalent projectors for virtual element methods

Ahmad, Bashir; Alsaedi, Ahmed; Brezzi, Franco; Marini, L. D.; Russo, A.

doi:10.1016/j.camwa.2013.05.015

Cited by 530 publications

(634 citation statements)

References 28 publications

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“…Remark 2 (Extension to three spatial dimensions) Implementing the virtual element method for problems in three spatial dimensions on polyhedral meshes requires the use of the so-called enhanced virtual element spaces introduced in [1]. In these modified local function spaces, it becomes possible to compute the local L 2 -orthogonal projection of virtual element functions onto the polynomial subspace on each mesh element and face.…”

Section: Applying the Boundary Conditionsmentioning

confidence: 99%

The virtual element method in 50 lines of MATLAB

Sutton

2016

Numer Algor

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We present a 50-line MATLAB implementation of the lowest order virtual element method for the two-dimensional Poisson problem on general polygonal meshes. The matrix formulation of the method is discussed, along with the structure of the overall algorithm for computing with a virtual element method. The purpose of this software is primarily educational, to demonstrate how the key components of the method can be translated into code.

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Section: Applying the Boundary Conditionsmentioning

confidence: 99%

The virtual element method in 50 lines of MATLAB

Sutton

2016

Numer Algor

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“…First introduced in [4] and extended in [5,6,3,21,2,26], the Virtual Element Method allows the use of quite general non-degenerate and star-shaped polygons to mesh the spatial domain, even including the possibility of straight angles. In the present framework, we take advantage from this flexibility to easily build a mesh which, on each fracture, is locally or globally conforming to the traces.…”

Section: The Discrete Dfn Problemmentioning

confidence: 99%

“…In this work we recall some results concerning the application of the Virtual Element Method [4,3,2] to the steady state simulation of the flow in DFNs [1,22,27,30,24,32,23,15,16,17,18,19,9,10,8,13]. In this approach we can exploit the flexibility of VEM in order to tackle the geometrical complexity.…”

Section: Introductionmentioning

confidence: 99%

The Virtual Element Method for Discrete Fracture Network Flow and Transport Simulations

Benedetto¹,

Berrone²,

Borio³

et al. 2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. We discuss several issues concerning the application of the Virtual Element Method (VEM) to the flow in fractured media modeled by the Discrete Fracture Network (DFN) model. Due to the stochastic nature of the computational domains, several geometrical complexities make the computations very challenging. The geometrical flexibility provided by the Virtual Element Method can be exploited to mutually couple local problems, either by resorting to a Mortar approach, or by allowing for the global conformity of the local meshes, while keeping the computational cost under control. We describe these two approaches in detail and we test them on a realistic test case, showing the viability of the two approaches.

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“…(See the treatment of the original virtual element method in [10].) The first goal of our paper is to extend some basic finite element estimates to the virtual elements introduced in [1], under the shape regularity assumptions that can be found for example in [1,4,8]. The main tool is a discrete norm for virtual element functions that plays the role of the L norm in the analysis of standard Lagrange finite element functions and which can be controlled by standard shape regularity arguments.…”

Section: Introductionmentioning

confidence: 99%