Lowest order virtual element approximations for transient Stokes problem on polygonal meshes

Verma, Nitesh; Kumar, Sarvesh

doi:10.1007/s10092-021-00440-7

Cited by 8 publications

(4 citation statements)

References 46 publications

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“…By extending the analysis of [12,29], we have discussed and analyzed the lowest order conforming virtual element methods for the approximation of coupled poroelasticity and ADR equations. The major contributions of this article are: showing the well-posedness of fully discrete schemes and establishing the optimal a-priori error estimates for all the variables that naturally appeared in the weak formulation.…”

Section: Discussionmentioning

confidence: 99%

“…The approximation space for displacement is V h , for total pressure, Z h , and for pressure and concentration is Q h (K). We have the following degrees of freedom depending on the corresponding spaces (refer [1,4,29] for details on unisolvance) as, for displacement: the values at all vertices of the element K, and value of v s • n e K at mid point on each edge e ∈ ∂K; for total volumetric stress: value of ψ at any point in K; and for pressure, or concentration: the values of q f at vertices of the element K.…”

Section: Virtual Element Discretizationsmentioning

confidence: 99%

“…Also, the discrete inf-sup condition hold on (V h , Z h ) (refer [7,8,29]). By introducing the adequate discrete spaces associated with velocity, pressure, total volumetric stress, and concentrations, the fully discrete formulation is described in the next subsection.…”

Section: Virtual Element Discretizationsmentioning

confidence: 99%

See 2 more Smart Citations

Virtual Element Approximations for Two Species Model of the Advection-Diffusion-Reaction in Poroelastic Media

Verma

Kumar

2022

Mathematical Modelling and Analysis

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This paper proposes virtual element methods for approximating the mathematical model consisting of coupled poroelastic and Advection-Diffusion-Reaction (ADR) equations. The space discretization relies on virtual element spaces containing piecewise linear polynomials as well as non-polynomials for displacement, pressure and concentrations, and piecewise constant for total pressure; a backwardEuler scheme is employed for the approximation of time derivative. Using standard techniques of explicit schemes, the well-posedness of the resultant fully discrete scheme is derived. Moreover, under certain regularity assumptions on the mesh, optimal apriori error estimates are established by introducing suitable projection operators. Several numerical experiments are presented to validate the theoretical convergence rate and exhibit the proposed scheme’s performance.

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Section: Discussionmentioning

confidence: 99%

Section: Virtual Element Discretizationsmentioning

confidence: 99%

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Virtual Element Approximations for Two Species Model of the Advection-Diffusion-Reaction in Poroelastic Media

Verma

Kumar

2022

Mathematical Modelling and Analysis

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“…In particular, we employ here the VEM for NS equations introduced in [6]. Other formulations (of mixed, discontinuous, nonconforming, and other types) for NS include [7,22,36,46,48], whereas for the PNP system a VEM scheme has been recently proposed in [37]. The present method also follows other VEM formulations for Stokes flows from [5,10,12,49].…”

Section: Scopementioning

confidence: 99%

Optimal error estimates of coupled and divergence-free virtual element methods for the Poisson--Nernst--Planck/Navier--Stokes equations

Dehghan¹,

Gharibi²,

Ruiz–Baier³

2022

Preprint

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In this article, we propose and analyze a fully coupled, nonlinear, and energy-stable virtual element method (VEM) for solving the coupled Poisson-Nernst-Planck (PNP) and Navier-Stokes (NS) equations modeling microfluidic and electrochemical systems (diffuse transport of charged species within incompressible fluids coupled through electrostatic forces). A mixed VEM is employed to discretize the NS equations whereas classical VEM in primal form is used to discretize the PNP equations. The stability, existence and uniqueness of solution of the associated VEM are proved by fixed point theory. Global mass conservation and electric energy decay of the scheme are also proved. Also, we obtain unconditionally optimal error estimates for both the electrostatic potential and ionic concentrations of PNP equations in the H 1 -norm, as well as for the velocity and pressure of NS equations in the H 1 -and L 2 -norms, respectively. Finally, several numerical experiments are presented to support the theoretical analysis of convergence and to illustrate the satisfactory performance of the method in simulating the onset of electrokinetic instabilities in ionic fluids, and studying how they are influenced by different values of ion concentration and applied voltage. These tests relate to applications in the desalination of water.Keywords Coupled Poisson-Nernst-Planck/Navier-Stokes equations • mixed virtual element method • optimal convergence • charged species transport • electrokinetic instability • water desalination.

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Optimal Error Estimates of Coupled and Divergence-Free Virtual Element Methods for the Poisson–Nernst–Planck/Navier–Stokes Equations and Applications in Electrochemical Systems

2023

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Lowest order virtual element approximations for transient Stokes problem on polygonal meshes

Cited by 8 publications

References 46 publications

Virtual Element Approximations for Two Species Model of the Advection-Diffusion-Reaction in Poroelastic Media

Virtual Element Approximations for Two Species Model of the Advection-Diffusion-Reaction in Poroelastic Media

Optimal error estimates of coupled and divergence-free virtual element methods for the Poisson--Nernst--Planck/Navier--Stokes equations

Optimal Error Estimates of Coupled and Divergence-Free Virtual Element Methods for the Poisson–Nernst–Planck/Navier–Stokes Equations and Applications in Electrochemical Systems

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