A Mixed Virtual Element Method for The Boussinesq Problem on Polygonal meshes

Gatica, Gabriel N.; Munar, Mauricio; Sequeira, Filánder A.

doi:10.4208/jcm.2001-m2019-0187

Cited by 11 publications

(4 citation statements)

References 31 publications

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“…In this section we present a numerical experiment in order to illustrate the performance of the mixed virtual element scheme (5.6) employing the Newton's iteration introduced in (5.8). It allows us to validate the operators introduced in Section 4, together with the numerical experiments presented in recent papers about mixed-VEM schemes, which utilized our implementation approach (see [15,17,16,26,25,32]). We begin by recalling from (5.10) that N stands for the total number of degrees of freedom (unknowns) of (5.8).…”

Section: Numerical Resultsmentioning

confidence: 99%

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Some aspects on the computational implementation of diverse terms arising in mixed virtual element formulations

Sequeira¹,

Guillén-Oviedo²

2020

Preprint

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In the present paper we describe the computational implementation of some integral terms that arise from mixed virtual element methods (mixed-VEM) in two-dimensional pseudostress-velocity formulations. The implementation presented here consider any polynomial degree k ≥ 0 in a natural way by building several local matrices of small size through the matrix multiplication and the Kronecker product. In particular, we apply the foregoing mentioned matrices to the Navier-Stokes equations with Dirichlet boundary conditions, whose mixed-VEM formulation was originally proposed and analyzed in a recent work using virtual element subspaces for H(div) and H 1 , simultaneously. In addition, an algorithm is proposed for the assembly of the associated global linear system for the Newton's iteration. Finally, we present a numerical example in order to illustrate the performance of the mixed-VEM scheme and confirming the expected theoretical convergence rates.

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Section: Numerical Resultsmentioning

confidence: 99%

“…We begin by describing in detail a way to assemble diverse local terms arising in mixed-primal virtual element formulations as in [15,17,16,27,26,25]. In particular, we are interested in those coming from the scheme proposed in [27] for the Navier-Stokes system, which is recalling in Section 5.2 below.…”

Section: Local Discrete Operators Arising From Vem Schemesmentioning

confidence: 99%

Some aspects on the computational implementation of diverse terms arising in mixed virtual element formulations

Sequeira¹,

Guillén-Oviedo²

2020

Preprint

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“…It has been used in a variety of fields, such as discrete fracture network simulation, incompressible miscible displacements in porous media, resistive magnetohydrodynamics and polycrystal composite materials. Since the original introduction of [3], various problems have been solved by the virtual element method so far, for example [1,2,5,10,18,26,37]. The VEM can handle very general polygonal elements with geometrical hanging nodes, because we just treat the hanging nodes as new nodes.…”

Section: Introductionmentioning

confidence: 99%

Adaptive Virtual Element Method for Convection Dominated Diffusion Equations

Zhou

2023

JCM

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In this paper, a robust residual-based a posteriori estimate is discussed for the Streamline Upwind/Petrov Galerkin (SUPG) virtual element method (VEM) discretization of convection dominated diffusion equation. A global upper bound and a local lower bound for the a posteriori error estimates are derived in the natural SUPG norm, where the global upper estimate relies on some hypotheses about the interpolation errors and SUPG virtual element discretization errors. Based on the Dörfler's marking strategy, adaptive VEM algorithm drived by the error estimators is used to solve the problem on general polygonal meshes. Numerical experiments show the robustness of the a posteriori error estimates.

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“…Several virtual element methods based on conforming and non-conforming schemes have been developed to solve a wide variety of problems in Solid and Fluid Mechanics, for example [4][5][6]9,11,12,14,19,25,27,30,42,46,47]. Moreover, the VEM for thin structures has been developed in [16,24,29,30,44,45], whereas VEM for nonlinear problems have been introduced in [3,15,26,35,36,50] In this paper, we analyze a conforming 1 Virtual Element Method to approximate the isolated solutions of the von Kármán equations. We consider a variational formulation in terms of the transverse displacement and the Airy stress function, which contains bilinear and trilinear forms.…”

Section: Introductionmentioning

confidence: 99%

A virtual element method for the von Kármán equations

2021

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In this article we propose and analyze a Virtual Element Method (VEM) to approximate the isolated solutions of the von Kármán equations, which describe the deformation of very thin elastic plates. We consider a variational formulation in terms of two variables: the transverse displacement of the plate and the Airy stress function. The VEM scheme is conforming in H2 for both variables and has the advantages of supporting general polygonal meshes and is simple in terms of coding aspects. We prove that the discrete problem is well posed for h small enough and optimal error estimates are obtained. Finally, numerical experiments are reported illustrating the behavior of the virtual scheme on different families of meshes.

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A Mixed Virtual Element Method for The Boussinesq Problem on Polygonal meshes

Cited by 11 publications

References 31 publications

Some aspects on the computational implementation of diverse terms arising in mixed virtual element formulations

Some aspects on the computational implementation of diverse terms arising in mixed virtual element formulations

Adaptive Virtual Element Method for Convection Dominated Diffusion Equations

A virtual element method for the von Kármán equations

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