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

Adak, Dibyendu; Mora, David; Natarajan, Sundararajan; Silgado, Alberth

doi:10.1051/m2an/2021058

Cited by 16 publications

(4 citation statements)

References 41 publications

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“…Proof. The estimations of u − R h (u) in • 2,Ω and • 1,Ω norms can be proved following analogous arguments as [1]. Now, we proceed to prove the estimation of u − R h (u) in • 0,Ω -norm.…”

Section: Convergence Analysis Of the Discrete Schemementioning

confidence: 86%

“…Following [1,39], it can be prove that the discrete bilinear form A h (•, •) is coercive and for any function ω ∈ H 2 * (Ω), A(ω, •) is continuous on Z h . By using the Lax-Milgram Theorem, we can conclude that (4.44) has unique solution.…”

Section: Convergence Analysis Of the Discrete Schemementioning

confidence: 99%

“…The VEM also permits to easily implement highly regular conforming discrete spaces [18,22] which make the method very feasible to solve various fourth-order problems [8,35,14,34,36]. Regarding VEM for time dependent problems, we mention the following works [2,1,4,6,9,39,38,40].…”

Section: Introductionmentioning

confidence: 99%

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Convergence Analysis of Virtual Element Method for Nonlinear Nonlocal Dynamic Plate Equation

Adak¹,

Mora²,

Natarajan³

2022

Preprint

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In this article, we have considered a nonlinear nonlocal time dependent fourth order equation demonstrating the deformation of a thin and narrow rectangular plate. We propose C 1 conforming virtual element method (VEM) of arbitrary order, k ≥ 2, to approximate the model problem numerically. We employ VEM to discretize the space variable and fully implicit scheme for temporal variable. Well-posedness of the fully discrete scheme is proved under certain conditions on the physical parameters, and we derive optimal order of convergence in both space and time variable. Finally, numerical experiments are presented to illustrate the behaviour of the proposed numerical scheme.

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Section: Convergence Analysis Of the Discrete Schemementioning

confidence: 86%

Section: Convergence Analysis Of the Discrete Schemementioning

confidence: 99%

See 1 more Smart Citation

Convergence Analysis of Virtual Element Method for Nonlinear Nonlocal Dynamic Plate Equation

Adak¹,

Mora²,

Natarajan³

2022

Preprint

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“…Kim et al (2007) has employed meshfree collocation method to obtain the solution of the NS equations in stream function-vorticity form. Recently, Adak et al (2021) has used virtual element method to propose a solution for the time-dependent NS equations while El-Amrani et al has presented enriched Galerkin-characteristics finite element methods to obtain the promising results for the NS equations.…”

Section: Introductionmentioning

confidence: 99%

On the BEM solution of convection–diffusion type equations involving variable convective coefficients

Fendoğlu,

Bozkaya,

Tezer-Sezgin

2024

Fluid Dyn. Res.

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In this study, numerical solutions of the unsteady convection–diffusion type equations of variable convective coefficients are investigated by two effective techniques alternative to direct boundary element method (BEM). That is, the domain boundary element and the dual reciprocity BEMs with the fundamental solution of convection–diffusion equation are used in space to transform the governing differential equations into equivalent integral equations while a backward finite difference scheme is utilized in time discretization. The fruitfulness of these combined techniques are shown by the implementation of the techniques for some widely investigated fluid dynamics problems. In fact, the lid-driven cavity flow is solved, and precise results are attained as benchmarks for assessing the accuracy of the aforementioned methods. Further, the application of the methods is extended for the natural convection flow which is governed mainly by the convection–diffusion type equations accompanied by the velocity components as variable convective coefficients. The obtained numerical results reveal that the domain BEM which uses the fundamental solution of convection–diffusion equation captures the characteristics and physical advancement of the fluid flow quite well at various combined values of the problem physical parameters.

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Convergence Analysis of Virtual Element Method for Nonlinear Nonlocal Dynamic Plate Equation

2022

View full text Add to dashboard Cite

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

Cited by 16 publications

References 41 publications

Convergence Analysis of Virtual Element Method for Nonlinear Nonlocal Dynamic Plate Equation

Convergence Analysis of Virtual Element Method for Nonlinear Nonlocal Dynamic Plate Equation

On the BEM solution of convection–diffusion type equations involving variable convective coefficients

Convergence Analysis of Virtual Element Method for Nonlinear Nonlocal Dynamic Plate Equation

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