Some Estimates for Virtual Element Methods

Brenner, Susanne C.; Guan, Qingguang; Sung, Li-yeng

doi:10.1515/cmam-2017-0008

Cited by 152 publications

(111 citation statements)

References 24 publications

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“…Once we defined the projection Π ∇ k , and checked that it is computable, in practice, for every v ∈ V k (P), we can extend it, in the obvious way, to an element v in G{V k (P)} by setting (25) Π ∇ k v := Π ∇ kv . Then we can follow the usual track of Virtual Elements, setting, for u and v in G{V k (P)}:…”

Section: The Local Vem Stiffness Matricesmentioning

confidence: 99%

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The Virtual Element Method with curved edges

2019

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In this paper we initiate the investigation of Virtual Elements with curved faces. We consider the case of a fixed curved boundary in two dimensions, as it happens in the approximation of problems posed on a curved domain or with a curved interface. While an approximation of the domain with polygons leads, for degree of accuracy k≥2, to a sub-optimal rate of convergence, we show (both theoretically and numerically) that the proposed curved VEM lead to an optimal rate of convergence.

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Section: The Local Vem Stiffness Matricesmentioning

confidence: 99%

“…Moreover, always for simplicity, we fix our attention on the dofi-dofi stabilization (30). Under our assumptions on the mesh the validity of (27), that is, estimates from below, can be easily proved with the techniques used in [15], [25]. Therefore we only have to deal with estimates from above.…”

Section: Interpolation Estimatesmentioning

confidence: 99%

The Virtual Element Method with curved edges

2019

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“…where is a uniform positive constant. We remark that the hypotheses (A1), (A2) and (A3), though not too restrictive in many practical cases, can be further relaxed, as investigated in [18,25,26,30].…”

Section: Basic Notations and Mesh Assumptionsmentioning

confidence: 99%

The Stokes complex for Virtual Elements in three dimensions

Veiga

Dassi

Vacca

2020

Math. Models Methods Appl. Sci.

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The present paper has two objectives. On one side, we develop and test numerically divergence free Virtual Elements in three dimensions, for variable "polynomial" order. These are the natural extension of the two-dimensional divergence free VEM elements, with some modification that allows for a better computational efficiency. We test the element's performance both for the Stokes and (diffusion dominated) Navier-Stokes equation. The second, and perhaps main, motivation is to show that our scheme, also in three dimensions, enjoys an underlying discrete Stokes complex structure. We build a pair of virtual discrete spaces based on general polytopal partitions, the first one being scalar and the second one being vector valued, such that when coupled with our velocity and pressure spaces, yield a discrete Stokes complex. *

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“…However, the available information on them is sufficient to form the stiffness matrix and the right-hand side of the discretized problem. For the analysis of the VEM technique for the basic second-order elliptic problems, we refer to [10,8,14].…”

Section: Introductionmentioning

confidence: 99%

A dual hybrid virtual element method for plane elasticity problems

et al. 2020

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A dual hybrid Virtual Element scheme for plane linear elastic problems is presented and analysed. In particular, stability and convergence results have been established. The method, which is first order convergent, has been numerically tested on two benchmarks with closed form solution, and on a typical microelectromechanical system. The numerical outcomes have proved that the dual hybrid scheme represents a valid alternative to the more classical low-order displacementbased Virtual Element Method.

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Some Estimates for Virtual Element Methods

Abstract: We present novel techniques for obtaining the basic estimates of virtual element methods in terms of the shape regularity of polygonal/polyhedral meshes. We also derive new error estimates for the Poisson problem in two and three dimensions.

Cited by 152 publications

References 24 publications

The Virtual Element Method with curved edges

The Virtual Element Method with curved edges

The Stokes complex for Virtual Elements in three dimensions

A dual hybrid virtual element method for plane elasticity problems

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