The NonConforming Virtual Element Method for the Stokes Equations

Cangiani, Andrea; Gyrya, Vitaliy; Manzini, Gianmarco

doi:10.1137/15m1049531

Cited by 161 publications

(71 citation statements)

References 37 publications

(59 reference statements)

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“…The r-consistency property follows by noting that the stability term in (26) is zero when one of its entries is a polynomial of degree r as Π ∇,K r is a polynomial-preserving operator. The stability property is easily established by applying (9) to definition (26) and setting α * = min(σ * , 1) and α * = max(σ * , 1), where σ * and σ * are the constants defined in (27).…”

Section: Construction Of the Bilinear Formmentioning

confidence: 99%

The conforming virtual element method for polyharmonic problems

Antonietti

Manzini

Verani

2020

Computers & Mathematics with Applications

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In this work, we exploit the capability of virtual element methods in accommodating approximation spaces featuring high-order continuity to numerically approximate differential problems of the form ∆ p u = f , p ≥ 1. More specifically, we develop and analyze the conforming virtual element method for the numerical approximation of polyharmonic boundary value problems, and prove an abstract result that states the convergence of the method in the energy norm.

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Section: Construction Of the Bilinear Formmentioning

confidence: 99%

The conforming virtual element method for polyharmonic problems

Antonietti

Manzini

Verani

2020

Computers & Mathematics with Applications

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“…Proof. The convergence analysis of the error u − u h in the H 1 norm begins by considering v h = D t ρ 2 (t) in (27), which follows as:…”

Section: 2mentioning

confidence: 99%

“…Due to its versatility, the method has been applied to a wide variety of problems. Some of them include: the Stokes equation [19], the Plate bending equation [20], linear elliptic, parabolic, and hyperbolic equations [16,17,18,21], linear and nonlinear elasticity problems [22,23], the convection diffusion equation with small diffusion [24], eigenvalue problems [25,26], nonconforming VEM for Stokes and elliptic equations [27,28,29].Recently an open source C++ library has been developed by [30]. Sutton [31] developed a 50-line MATLAB implementation for the lowest order VEM for the two-dimensional Poisson's problem.…”

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confidence: 99%

Virtual element method for semilinear sine–Gordon equation over polygonal mesh using product approximation technique

Adak

Natarajan

2020

Mathematics and Computers in Simulation

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In this paper, we employ the linear virtual element spaces to discretize the semilinear sine-Gordon equation in two dimensions. The salient features of the virtual element method (VEM) are: (a) it does not require explicit form of the shape functions to construct the nonlinear and the bilinear terms, and (b) relaxes the constraint on the mesh topology by allowing the domain to be discretized with general polygons consisting of both convex and concave elements, and (c) easy mesh refinements (hanging nodes and interfaces are allowed ). The nonlinear source term is discretized by employing the product approximation technique and for temporal discretization, the Crank-Nicolson scheme is used. The resulting nonlinear equations are solved using the Newton's method. We derive a priori error estimations in L 2 and H 1 norms. The convergence properties and the accuracy of the virtual element method for the solution of the sine-Gordon equation are demonstrated with academic numerical experiments. studies focused on undamped sine-Gordon equation. The damped sine-Gordon equation demands more sophisticated numerical techniques. In this direction, the following articles are presented in the literature: Nakajima et al. [6] proposed an efficient numerical technique for the solution of the sine-Gordon equation considering dimensionless loss factors and unit less normalized bias. Recently, a dual reciprocity boundary element method [4] and the radial basis function method [7] have been introduced for the above mentioned model problem. However, most of these techniques require the domain to be discretized with simplex elements such as triangles/quadrilaterals.The introduction of elements with arbitrary number of sides has revolutionized the simulation technique. Elements with arbitrary number of sides offer greater flexibility in meshing and are less sensitive to mesh distortion [8,9]. Among the different approaches based on polygonal elements, such as the scaled boundary finite element method [10], the polygonal FEM [8,11,12], the strain smoothing techniques [13,14,15], the virtual element method (VEM) [16,17,18] has an edge. The VEM is a recently developed framework, which provides FEM-like approximations over arbitrary polygonal meshes. Some of the salient features of the VEM are: (a) it does not require an explicit form of the shape functions to compute the bilinear form; (b) it can be applied to convex, and concave elements, i.e., star convexity is not required; (c) easy mesh refinements (hanging nodes and interfaces are allowed ) and (d) the degrees of freedom associated with the VEM space, provides sufficient information to compute the discrete bilinear form. Due to its versatility, the method has been applied to a wide variety of problems. Some of them include: the Stokes equation [19], the Plate bending equation [20], linear elliptic, parabolic, and hyperbolic equations [16,17,18,21], linear and nonlinear elasticity problems [22,23], the convection diffusion equation with small diffusion [24], eigenvalue problems [25,26], ...

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“…An advantage of the proposed family of Virtual Elements is that, without the need of a high minimal polynomial degree as it happens in conforming FEM, it is able to yield a discrete divergence-free (conforming) velocity solution, which can be an interesting asset as explored for Finite Elements in [44,46,53,48,45]. For a wider look in the literature, other VEM for Stokes-type problems can be found in [50,29,28,42,31,35] while different polygonal methods for the same problem in [49,33,24,38].…”

Section: Introductionmentioning

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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The NonConforming Virtual Element Method for the Stokes Equations

Cited by 161 publications

References 37 publications

The conforming virtual element method for polyharmonic problems

The conforming virtual element method for polyharmonic problems

Virtual element method for semilinear sine–Gordon equation over polygonal mesh using product approximation technique

The Stokes complex for Virtual Elements in three dimensions

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