A mixed virtual element method for the pseudostress–velocity formulation of the Stokes problem

Cáceres, Ernesto; Gatica, Gabriel N.

doi:10.1093/imanum/drw002

Cited by 105 publications

(93 citation statements)

References 33 publications

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“…We first establish (23). We begin by introducing the following well-posed auxiliary problem: Find ( w, γ) ∈ X such that (25) a…”

Section: Proposition 44mentioning

confidence: 99%

“…Next, choosing v := (w − w h ) and τ = (γ − γ h ) in (25), so that ∇v + τ = θ − θ h , and then adding and subtracting the term ∇ w…”

Section: Proposition 44mentioning

confidence: 99%

“…Indeed, by avoiding the explicit construction of the local basis functions, the VEM can easily handle general polygons/polyhedrons without complex integrations on the element (see [9] for details on the coding aspects of the method). The Virtual Element Method has been applied successfully in a large range of problems, see for instance [1,2,7,8,9,12,15,16,17,20,23,25,28,35,39,40,41,47,48].…”

Section: Introductionmentioning

confidence: 99%

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Virtual elements for a shear-deflection formulation of Reissner–Mindlin plates

Veiga

Mora²,

Rivera

2018

Math. Comp.

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Abstract. We present a virtual element method for the Reissner-Mindlin plate bending problem which uses shear strain and deflection as discrete variables without the need of any reduction operator. The proposed method is conforming in [H 1 (Ω)] 2 × H 2 (Ω) and has the advantages of using general polygonal meshes and yielding a direct approximation of the shear strains. The rotations are then obtained by a simple postprocess from the shear strain and deflection. We prove convergence estimates with involved constants that are uniform in the thickness t of the plate. Finally, we report numerical experiments which allow us to assess the performance of the method.

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“…We first establish (23). We begin by introducing the following well-posed auxiliary problem: Find ( w, γ) ∈ X such that (25) a…”

Section: Proposition 44mentioning

confidence: 99%

“…Next, choosing v := (w − w h ) and τ = (γ − γ h ) in (25), so that ∇v + τ = θ − θ h , and then adding and subtracting the term ∇ w…”

Section: Proposition 44mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Virtual elements for a shear-deflection formulation of Reissner–Mindlin plates

Veiga

Mora²,

Rivera

2018

Math. Comp.

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“…The VEM framework has already been applied to the streamline formulation of the Stokes equation in [4], a pseudo-stress velocity formulation can be found in [21], and a divergence free virtual approach can be found in [13]. Among the recent developments in discontinuous Galerkin methods, it is worth mentioning [20], the Hybridized Discontinuous Galerkin [24,25], and, concerning polygonal meshes, the Hybrid High Order method [1] and the Weak Galerkin method [38].…”

mentioning

confidence: 99%

The NonConforming Virtual Element Method for the Stokes Equations

Cangiani¹,

Gyrya²,

Manzini³

2016

SIAM J. Numer. Anal.

161

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Abstract. We present the non-conforming Virtual Element Method (VEM) for the numerical approximation of velocity and pressure in the steady Stokes problem. The pressure is approximated using discontinuous piecewise polynomials, while each component of the velocity is approximated using the nonconforming virtual element space. On each mesh element the local virtual space contains the space of polynomials of up to a given degree, plus suitable non-polynomial functions. The virtual element functions are implicitly defined as the solution of local Poisson problems with polynomial Neumann boundary conditions. As typical in VEM approaches, the explicit evaluation of the non-polynomial functions is not required. This approach makes it possible to construct nonconforming (virtual) spaces for any polynomial degree regardless of the parity, for two-and three-dimensional problems, and for meshes with very general polygonal and polyhedral elements. We show that the non-conforming VEM is inf-sup stable and establish optimal a priori error estimates for the velocity and pressure approximations. Numerical examples confirm the convergence analysis and the effectiveness of the method in providing high-order accurate approximations.

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“…We here limit ourselves in citing [3,10,12,13,[18][19][20][21]25,26,28,33,34,44,45,51] and refer to [15] for a more complete overview of the literature.…”

mentioning

confidence: 99%

Exponential convergence of the hp virtual element method in presence of corner singularities

et al. 2017

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In the present work, we analyze the hp version of virtual element methods for the 2D Poisson problem. We prove exponential convergence of the energy error employing sequences of polygonal meshes geometrically refined, thus extending the classical choices for the decomposition in the hp finite element framework to very general decomposition of the domain. A new stabilization for the discrete bilinear form with explicit bounds in h and p is introduced. Numerical experiments validate the theoretical results. We also exhibit a numerical comparison between hp virtual elements and hp finite elements. Mathematics Subject Classification

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A mixed virtual element method for the pseudostress–velocity formulation of the Stokes problem

Cited by 105 publications

References 33 publications

Virtual elements for a shear-deflection formulation of Reissner–Mindlin plates

Virtual elements for a shear-deflection formulation of Reissner–Mindlin plates

The NonConforming Virtual Element Method for the Stokes Equations

Exponential convergence of the hp virtual element method in presence of corner singularities

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