Stabilized virtual element methods for the unsteady incompressible Navier–Stokes equations

Irisarri, Diego; Hauke, Guillermo

doi:10.1007/s10092-019-0332-5

Cited by 34 publications

(5 citation statements)

References 37 publications

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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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Section: Introductionmentioning

confidence: 99%

p- and hp- virtual elements for the Stokes problem

2021

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“…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 [7] for details on the coding aspects of the method). The VEM has been applied successfully for problems in fluid mechanics; see for instance [1,10,18,24,26,30,34,33,17,39,41], where Stokes, Brinkman, Stokes-Darcy and Navier-Stokes equations have been recently developed.…”

Section: Introductionmentioning

confidence: 99%

A virtual element discretization for the time dependent Navier–Stokes equations in stream-function formulation

et al. 2021

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In this work, a new Virtual Element Method (VEM) of arbitrary order $k \geq 2$ for the time dependent Navier-Stokes equations in stream-function form is proposed and analyzed. Using suitable projection operators, the bilinear and trilinear terms are discretized by only using the proposed degrees of freedom associated with the virtual space. Under certain assumptions on the computational domain, error estimations are derived and shown that the method is optimally convergent in both space and time variables. Finally, to justify the theoretical analysis, four benchmark examples are examined numerically.

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“…An important area that is left out is the development of high order spectral volume and spectral difference methods advanced by Kannan et al [ 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 ] and Sun et al [ 24 ]. In recent years, the weak Galerkin method [ 25 ] and virtual element method [ 26 , 27 , 28 ] have also made great contributions to solve the Navier–Stokes equations. Chen et al in [ 29 ] proposed a dimension splitting method for the 3D steady Navier–Stokes equations and in [ 30 ], proposed a dimension splitting and characteristic projection method for the 3D time-dependent Navier–Stokes equations, giving some numerical examples to verify the effectiveness of the algorithm.…”

Section: Introductionmentioning

confidence: 99%

Finite Element Iterative Methods for the 3D Steady Navier--Stokes Equations

2021

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In this work, a finite element (FE) method is discussed for the 3D steady Navier–Stokes equations by using the finite element pair Xh×Mh. The method consists of transmitting the finite element solution (uh,ph) of the 3D steady Navier–Stokes equations into the finite element solution pairs (uhn,phn) based on the finite element space pair Xh×Mh of the 3D steady linearized Navier–Stokes equations by using the Stokes, Newton and Oseen iterative methods, where the finite element space pair Xh×Mh satisfies the discrete inf-sup condition in a 3D domain Ω. Here, we present the weak formulations of the FE method for solving the 3D steady Stokes, Newton and Oseen iterative equations, provide the existence and uniqueness of the FE solution (uhn,phn) of the 3D steady Stokes, Newton and Oseen iterative equations, and deduce the convergence with respect to (σ,h) of the FE solution (uhn,phn) to the exact solution (u,p) of the 3D steady Navier–Stokes equations in the H1−L2 norm. Finally, we also give the convergence order with respect to (σ,h) of the FE velocity uhn to the exact velocity u of the 3D steady Navier–Stokes equations in the L2 norm.

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Stabilized virtual element methods for the unsteady incompressible Navier–Stokes equations

Cited by 34 publications

References 37 publications

p- and hp- virtual elements for the Stokes problem

p- and hp- virtual elements for the Stokes problem

A virtual element discretization for the time dependent Navier–Stokes equations in stream-function formulation

Finite Element Iterative Methods for the 3D Steady Navier--Stokes Equations

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