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Volume 14, Issue 1
Backward Euler Schemes for the Kelvin-Voigt Viscoelastic Fluid Flow Model

A. K. Pany, S. K. Paikray, S. Padhy & A. K. Pani

Int. J. Numer. Anal. Mod., 14 (2017), pp. 126-151.

Published online: 2016-01

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  • Abstract

In this paper, we discuss the backward Euler method along with its linearized version for the Kelvin-Voigt viscoelastic fluid flow model with non zero forcing function, which is either independent of time or in $\rm{L}^∞(\rm{L^2})$. After deriving some bounds for the semidiscrete scheme, a priori estimates in Dirichlet norm for the fully discrete scheme are obtained, which are valid uniformly in time using a combination of discrete Gronwall's lemma and Stolz-Cesaro's classical result for sequences. Moreover, an existence of a discrete global attractor for the discrete problem is established. Further, optimal a priori error estimates are derived, whose bounds may depend exponentially in time. Under uniqueness condition, these estimates are shown to be uniform in time. Even when $\rm{f}$ = 0, the present result improves upon earlier result of Bajpai et al. (IJNAM,10 (2013), pp.481-507) in the sense that error bounds in this article depend on $1 / \sqrt{\kappa}$ as against $1 / \kappa^r$, $r \geq 1$. Finally, numerical experiments are conducted which confirm our theoretical findings.

  • AMS Subject Headings

65M60, 65M12, 65M15, 35D05, 35D10

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COPYRIGHT: © Global Science Press

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@Article{IJNAM-14-126, author = {}, title = {Backward Euler Schemes for the Kelvin-Voigt Viscoelastic Fluid Flow Model}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2016}, volume = {14}, number = {1}, pages = {126--151}, abstract = {

In this paper, we discuss the backward Euler method along with its linearized version for the Kelvin-Voigt viscoelastic fluid flow model with non zero forcing function, which is either independent of time or in $\rm{L}^∞(\rm{L^2})$. After deriving some bounds for the semidiscrete scheme, a priori estimates in Dirichlet norm for the fully discrete scheme are obtained, which are valid uniformly in time using a combination of discrete Gronwall's lemma and Stolz-Cesaro's classical result for sequences. Moreover, an existence of a discrete global attractor for the discrete problem is established. Further, optimal a priori error estimates are derived, whose bounds may depend exponentially in time. Under uniqueness condition, these estimates are shown to be uniform in time. Even when $\rm{f}$ = 0, the present result improves upon earlier result of Bajpai et al. (IJNAM,10 (2013), pp.481-507) in the sense that error bounds in this article depend on $1 / \sqrt{\kappa}$ as against $1 / \kappa^r$, $r \geq 1$. Finally, numerical experiments are conducted which confirm our theoretical findings.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/414.html} }
TY - JOUR T1 - Backward Euler Schemes for the Kelvin-Voigt Viscoelastic Fluid Flow Model JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 126 EP - 151 PY - 2016 DA - 2016/01 SN - 14 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/414.html KW - Viscoelastic fluids, Kelvin-Voigt model, a priori bounds, backward Euler method, discrete attractor, optimal error estimates, linearized backward Euler scheme, numerical experiments. AB -

In this paper, we discuss the backward Euler method along with its linearized version for the Kelvin-Voigt viscoelastic fluid flow model with non zero forcing function, which is either independent of time or in $\rm{L}^∞(\rm{L^2})$. After deriving some bounds for the semidiscrete scheme, a priori estimates in Dirichlet norm for the fully discrete scheme are obtained, which are valid uniformly in time using a combination of discrete Gronwall's lemma and Stolz-Cesaro's classical result for sequences. Moreover, an existence of a discrete global attractor for the discrete problem is established. Further, optimal a priori error estimates are derived, whose bounds may depend exponentially in time. Under uniqueness condition, these estimates are shown to be uniform in time. Even when $\rm{f}$ = 0, the present result improves upon earlier result of Bajpai et al. (IJNAM,10 (2013), pp.481-507) in the sense that error bounds in this article depend on $1 / \sqrt{\kappa}$ as against $1 / \kappa^r$, $r \geq 1$. Finally, numerical experiments are conducted which confirm our theoretical findings.

A. K. Pany, S. K. Paikray, S. Padhy & A. K. Pani. (1970). Backward Euler Schemes for the Kelvin-Voigt Viscoelastic Fluid Flow Model. International Journal of Numerical Analysis and Modeling. 14 (1). 126-151. doi:
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