On a two-grid finite element scheme combined with Crank–Nicolson method for the equations of motion arising in the Kelvin–Voigt model

Bajpai, Saumya; Nataraj, Neela

doi:10.1016/j.camwa.2014.07.011

Cited by 17 publications

(10 citation statements)

References 12 publications

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“…Thanks to (8) and Lemma 3.2, we have |b(e t n d t u n , u −1 (t n ), e t n d t u n )| ≤ 8 ||e t n d t u n || 2 1 + 4( 2 ) 3 c 4 0 ||u −1 (t n )|| 2 0 ||u −1 (t n )|| 2 1 ||e t n d t u n || 2 1 , |b(e t n u n−1 , d t u −1 (t n ), e t n d t u n ) + b(d t u −1 (t n ), e t n u n−1 , e t n d t u n )|…”

Section: Multilevel Space-time Finite Element Methods For the Kelvin-vmentioning

confidence: 99%

“…Theorem (see previous studies) Under the assumptions of (A1) and the domain

normalΩ

is a convex polygon, then problem has a uniqueness solution. Furthermore, it holds

‖ u_{t} (t) ‖_{2} + ‖ p (t) ‖_{1} + τ^{2} (t) ‖ u_{t} (t) ‖_{1} \leq c .

…”

Section: Preliminariesmentioning

confidence: 99%

“…5,6 Furthermore, based on the discrete Gronwall lemma and the Stolz-Cesaro classical result on sequences, Pani and his collaborators presented the optimal error estimates for the velocity and pressure in various norms. 5,7 Then, combined with the two grid methods, Bajpai and Nataraj 8 analyzed the Crank-Nicoson method for (1) and established the theoretical convergence analysis. For more numerical results about the Kelvin-Voigt model (1), we can refer to previous studies 6,[9][10][11] and the reference therein.…”

mentioning

confidence: 99%

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One‐level and multilevel space‐time finite element method for the viscoelastic Kelvin‐Voigt model

Zhang

Duan

2020

Math Meth Appl Sci

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In this paper, we consider the space‐time finite element method for the viscoelastic Kelvin‐Voigt model. Firstly, based on a priori estimates of spatial semidiscrete numerical solutions, stability and convergence results of one‐level space‐time finite element solutions in different norms are provided under some restrictions on the time step. Secondly, in order to improve the computational efficiency, multilevel space‐time finite element method is introduced. In the multilevel numerical scheme, the nonlinear Kelvin‐Voigt problem is just solved in the coarsest mesh, the Newton iteration is adopted to treat the nonlinear term, and a series of linear Kelvin‐Voigt problems are considered in successive shape conforming regular triangulations. The multilevel space‐time finite element approximate solution is shown to have a convergence rate of the same order as that of the one‐level space‐time finite element solutions of the nonlinear Kelvin‐Voigt problem on a fine mesh with some appropriate choices between the time steps and mesh sizes: kj∼kj−12,hj∼hj−13false/2. Finally, some numerical results are provided to verify the established theoretical findings. Compared with the standard Galerkin finite element method, from the view point of theoretical analysis, the space‐time finite element method has the same convergence orders for variables as the Galerkin method. However, from the view point of numerical simulations, the space‐time finite element method improves the computational accuracy without additional computational cost.

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Section: Multilevel Space-time Finite Element Methods For the Kelvin-vmentioning

confidence: 99%

“…Theorem (see previous studies) Under the assumptions of (A1) and the domain

normalΩ

is a convex polygon, then problem has a uniqueness solution. Furthermore, it holds

‖ u_{t} (t) ‖_{2} + ‖ p (t) ‖_{1} + τ^{2} (t) ‖ u_{t} (t) ‖_{1} \leq c .

…”

Section: Preliminariesmentioning

confidence: 99%

mentioning

confidence: 99%

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One‐level and multilevel space‐time finite element method for the viscoelastic Kelvin‐Voigt model

Zhang

Duan

2020

Math Meth Appl Sci

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“…Kuberry et al [15] have found that the Voigt regularization provides accurate reduced order models for Navier-Stokes equations for fluid flow. In [3,4], two fully discrete two-grid schemes have been applied to problem (1), where the second order accurate backward difference scheme and Crank-Nicolson scheme have been employed for time discretization. In [2], Bajpai et al have analyzed both backward Euler scheme and backward difference scheme for full discretization of the problem (1) with the forcing function f = 0.…”

Section: Introductionmentioning

confidence: 99%

A fully discrete finite element scheme for the Kelvin-Voigt model

Lü

Zhang

Huang

2019

Filomat

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In this paper, we study convergence of a fully discrete scheme for the two-dimensional nonstationary Kelvin-Voigt model. This scheme is based on a finite element approximation for space discretization and the Crank-Nicolson-type scheme for time discretization, which is a two step method. Moreover, we obtain error estimates of velocity and pressure. At last, the applicability and effectiveness of the present algorithm are illustrated by numerical experiments.

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“…Based on the the proof techniques of Ladyzenskaya [17], Oskolkov and his collaborators [18], [19], [21], [22] have discussed the existence of a unique global "almost " classical solution for the initial and boundary value problem (1.1)- (1.3) for various assumptions on the right-hand side function f and for all time t > 0. There is a considerable amount of literature devoted to the numerical approximations of Kelvin-Voigt fluid flow model, see [2]- [5], [16], [20], [25]- [28]. In [20], Oskolkov has applied the spectral Galerkin approximation to the problem (1.1)- (1.3) and has proved the convergence for t ≥ 0 with the assumption that the solution is asymptotically stable as t → ∞.…”

Section: Introductionmentioning

confidence: 99%

A priori error estimates of fully discrete finite element Galerkin method for Kelvin–Voigt viscoelastic fluid flow model

Bajpai

Pany

2019

Computers & Mathematics with Applications

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In this article, a finite element Galerkin method is applied to the Kelvin-Voigt viscoelastic fluid model, when its forcing function is in L ∞ (L 2 ). Some new a priori bounds for the velocity as well as for the pressure are derived which are independent of inverse powers of the retardation time κ. Optimal error estimates for the velocity in L ∞ (L 2 ) as well as in L ∞ (H 1 0 )-norms and for the pressure in L ∞ (L 2 )-norm of the semidiscrete method are discussed which hold uniformly with respect to κ as κ → 0 with the initial condition only in H 2 ∩ H 1 0 . Further, under uniqueness condition, these estimates are shown to be uniformly in time as t → ∞. For the complete discretization of the semidiscrete system, a first-order accurate backward Euler method is applied and fully discrete optimal error estimates are established. Finally, numerical experiments are conducted to verify the theoretical results. The results derived in this article are sharper than those derived earlier for finite element analysis of the Kelvin-Voigt fluid model in the sense that the error estimates in this article hold true uniformly even as κ → 0.Keywords: Kelvin-Voigt viscoelastic model, a priori estimates, semidiscrete finite element Galerkin method, fully discrete optimal error estimates, uniqueness condition.

show abstract

On a two-grid finite element scheme combined with Crank–Nicolson method for the equations of motion arising in the Kelvin–Voigt model

Cited by 17 publications

References 12 publications

One‐level and multilevel space‐time finite element method for the viscoelastic Kelvin‐Voigt model

One‐level and multilevel space‐time finite element method for the viscoelastic Kelvin‐Voigt model

A fully discrete finite element scheme for the Kelvin-Voigt model

A priori error estimates of fully discrete finite element Galerkin method for Kelvin–Voigt viscoelastic fluid flow model

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