The Stokes Complex for Virtual Elements with Application to Navier–Stokes Flows

Veiga, L. Beirão da; Mora, David; Vacca, Giuseppe

doi:10.1007/s10915-019-01049-3

Cited by 67 publications

(28 citation statements)

References 46 publications

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“…A theoretical study on possible pairs of spaces that satisfy the in f −sup condition for the proposed method on arbitrary polygonal/polyhedral meshes is under investigation. See [3,22,43] and [5,16,17,33] for a study of the Stokes problem on general meshes with the Hybrid High-Order and the Virtual Element methods, respectively.…”

Section: (Top-right)mentioning

confidence: 99%

A Polygonal Discontinuous Galerkin Formulation for Contact Mechanics in Fluid-Structure Interaction Problems

Zonca¹,

Antonietti²,

Vergara³

2021

CICP

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In this work, we propose a formulation based on the Polygonal Discontinuous Galerkin (PolyDG) method for contact mechanics that arises in fluid-structure interaction problems. In particular, we introduce a consistent penalization approach to treat the frictionless contact between immersed structures that undergo large displacements. The key feature of the method is that the contact condition can be naturally embedded in the PolyDG formulation, allowing to easily support polygonal/polyhedral meshes. The proposed approach introduced a fixed background mesh for the fluid problem overlapped by the structure grid that is able to move independently of the fluid one. To assess the validity of the proposed approach, we report the results of several numerical experiments obtained in the case of contact between flexible structures immersed in a fluid.

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Section: (Top-right)mentioning

confidence: 99%

A Polygonal Discontinuous Galerkin Formulation for Contact Mechanics in Fluid-Structure Interaction Problems

Zonca¹,

Antonietti²,

Vergara³

2021

CICP

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“…Once the proof for Σ h (P ) is given, the extension to the original space Σ h (P ) easily follows by employing standard techniques for VEM enhanced spaces (see [2] and Proposition 5.1 in [19]). Employing Theorem 6.1, given…”

Section: Proposition 32 the Dimension Of σ H (P ) Is Given Bymentioning

confidence: 99%

“…Remark 5.2. Since Proposition 4.2 yields an explicit characterization of Z h as curl Σ h , one could follow (61) and build an equivalent curl (discrete) formulation (see for instance Problem (77) in [19]). Such approach is less appealing in 3-d since the curl operator has a non trivial kernel and thus some stabilization or additional Lagrange multiplier would be needed in the formulation.…”

Section: The Discrete Problemmentioning

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 virtual element method (VEM) is an increasingly popular tool in the approximation to solutions of fluido-static and dynamic problems based on polygonal/polyhedral meshes. In particular we recall: the very first paper on low-order VEM for Stokes [2]; its high-order conforming [11] and nonconforming versions [20,34]; conforming [12] and nonconforming VEM for the Navier-Stokes equation [33]; mixed VEM for the pseudo-stress-velocity formulation of the Stokes problem [17]; mixed VEM for quasi-Newtonian flows [19]; mixed VEM for the Navier-Stokes equation [24]; other variants of the VEM for the Darcy problem [18,45,47]; analysis of the Stokes complex in the VEM framework [9,13]; a stabilized VEM for the unsteady incompressible Navier-Stokes equations [30]; implementation details [23].…”

Section: Introductionmentioning

confidence: 99%

p- and hp- virtual elements for the Stokes problem

2021

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We analyse the p- and hp-versions of the virtual element method (VEM) for the Stokes problem on polygonal domains. The key tool in the analysis is the existence of a bijection between Poisson-like and Stokes-like VE spaces for the velocities. This allows us to re-interpret the standard VEM for Stokes as a VEM, where the test and trial discrete velocities are sought in Poisson-like VE spaces. The upside of this fact is that we inherit from Beirão da Veiga et al. (Numer. Math. 138(3), 581–613, 2018) an explicit analysis of best interpolation results in VE spaces, as well as stabilization estimates that are explicit in terms of the degree of accuracy p of the method. We prove exponential convergence of the hp-VEM for Stokes problems with regular right-hand sides. We corroborate the theoretical estimates with numerical tests for both the p- and hp-versions of the method.

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The Stokes Complex for Virtual Elements with Application to Navier–Stokes Flows

Cited by 67 publications

References 46 publications

A Polygonal Discontinuous Galerkin Formulation for Contact Mechanics in Fluid-Structure Interaction Problems

A Polygonal Discontinuous Galerkin Formulation for Contact Mechanics in Fluid-Structure Interaction Problems

The Stokes complex for Virtual Elements in three dimensions

p- and hp- virtual elements for the Stokes problem

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