A modified nonlinear spectral Galerkin method for the equations of motion arising in the Kelvin–Voigt fluids

Pani, Amiya K.; Pany, Ambit Kumar; Damázio, Pedro D.; Yuan, Jinying

doi:10.1080/00036811.2013.841143

Cited by 12 publications

(14 citation statements)

References 25 publications

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“…Subsequently, Pani et a. [17] have applied a variant of nonlinear semidiscrete spectral Galerkin method and optimal error estimates are proved. It is, further, shown that a priori error estimates are valid uniformly in time under uniqueness assumption.…”

Section: Introductionmentioning

confidence: 99%

Optimal error estimates for semidiscrete Galerkin approximations to equations of motion described by Kelvin–Voigt viscoelastic fluid flow model

Pany

Bajpai

Pani

2016

Journal of Computational and Applied Mathematics

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In this paper, the finite element Galerkin method is applied to the equations of motion arising in the Kelvin-Voigt viscoelastic fluid flow model, when the forcing function is in L ∞ (L 2 ). Some a priori estimates for the exact solution, which are valid uniformly in time as t → ∞ and even uniformly in the retardation time κ an κ → 0, are derived. It is shown that the semidiscrete method admits a global attractor. Further, with the help of a priori bounds and Sobolev-Stokes projection, optimal error estimates for the velocity in L ∞ (L 2 ) and L ∞ (H 1 )-norms and for the pressure in L ∞ (L 2 )-norm are established. Since the constants involved in error estimates have an exponential growth in time, therefore, in the last part of the article, under certain uniqueness condition, the error bounds are established which are valid uniformly in time. Finally, some numerical experiments are conducted which confirm our theoretical findings.

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Section: Introductionmentioning

confidence: 99%

Optimal error estimates for semidiscrete Galerkin approximations to equations of motion described by Kelvin–Voigt viscoelastic fluid flow model

Pany

Bajpai

Pani

2016

Journal of Computational and Applied Mathematics

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“…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%

“…Theorem (see previous studies) Under the assumptions of (A1), (A2) and the domain

normalΩ

is a convex polygon, for all

t \geq 0

, the numerical solutions

u_{h}

and

p_{h}

of problem satisfy

\begin{matrix} false‖ u_{h} {false‖}_{1}^{2} + false‖ A_{h} u_{h} {false‖}_{0}^{2} + false‖ p_{h} {false‖}_{0}^{2} + false‖ u_{h t} {false‖}_{1}^{2} + τ false(t false) false‖ u_{h t} {false‖}_{1}^{2} + τ^{2} false(t false) false(false‖ A_{h} u_{h t} {false‖}_{0}^{2} + false‖ p_{h t} {false‖}_{1 h}^{2} false) \leq c, \\ \int_{0}^{t} false(false‖ u_{h t} {false‖}_{1}^{2} + τ false(s false) false‖ A_{h} u_{h t} {false‖}_{0}^{2} + τ false(s false) false‖ p_{h t} {false‖}_{1 h}^{2} + τ^{2} false(s false) false‖ u_{h t t} {false‖}_{1}^{2} + τ false(s false) false‖ u_{h t t} {false‖}_{0}^{2} false) d s \leq c, \end{matrix}

…”

Section: Galerkin Finite Element Approximationmentioning

confidence: 99%

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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“…Use of exponential weight in the energy arguments yields required a priori bounds. For the study of exponential decay properties of the exact as well as discrete solutions of the parabolic integrodifferential equations, in particular for Oldroyd model and Kelvin-Voigt model, we refer to [26][27][28][29][30][31]. When f = 0, the results provide improved exponential decay compared with the decay property shown in [5,7,8,18,22,32,33] for the problem in one space variable with a special case a (s) = (1 + s) p , 0 < p ≤ 1.…”

Section: A the Mathematical Modelmentioning

confidence: 99%

“…The results are new in the context of the present problem, when the forcing function is independent of time or it is in

L^{\infty} (0, \infty; L^{2} (Ω)) .

Use of exponential weight in the energy arguments yields required a priori bounds. For the study of exponential decay properties of the exact as well as discrete solutions of the parabolic integro‐differential equations, in particular for Oldroyd model and Kelvin–Voigt model, we refer to . When

f = 0,

the results provide improved exponential decay compared with the decay property shown in for the problem in one space variable with a special case

a (s) = {(1 + s)}^{p}, 0 < p \leq 1.

Applying Galerkin method in spatial direction while keeping time variable as continuous, a semidiscrete method is analyzed.…”

Section: Introductionmentioning

confidence: 99%

Finite element Galerkin approximations to a class of nonlinear and nonlocal parabolic problems

Sharma

Khebchareon

Sharma

et al. 2016

Numerical Methods Partial

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In this article, a finite element Galerkin method is applied to a general class of nonlinear and nonlocal parabolic problems. Based on an exponential weight function, new a priori bounds which are valid for uniform in time are derived. As a result, existence of an attractor is proved for the problem with nonhomogeneous right hand side which is independent of time. In particular, when the forcing function is zero or decays exponentially, it is shown that solution has exponential decay property which improves even earlier results in one dimensional problems. For the semidiscrete method, global existence of a unique discrete solution is derived and it is shown that the discrete problem has an attractor. Moreover, optimal error estimates are derived in both L 2 ( H 0 1 ( Ω ) ) and L ∞ ( H 0 1 ( Ω ) ) ‐norms with later estimate is a new result in this context. For completely discrete scheme, backward Euler method with its linearized version is discussed and existence of a unique discrete solution is established. Further, optimal estimates in ℓ 2 ( H 0 1 ( Ω ) ) ‐norm are proved for fully discrete schemes. Finally, several numerical experiments are conducted to confirm our theoretical findings. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1232–1264, 2016

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A modified nonlinear spectral Galerkin method for the equations of motion arising in the Kelvin–Voigt fluids

Cited by 12 publications

References 25 publications

Optimal error estimates for semidiscrete Galerkin approximations to equations of motion described by Kelvin–Voigt viscoelastic fluid flow model

Optimal error estimates for semidiscrete Galerkin approximations to equations of motion described by Kelvin–Voigt viscoelastic fluid flow model

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

Finite element Galerkin approximations to a class of nonlinear and nonlocal parabolic problems

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