Abstract
In this article, based on a second-order backward difference method, a completely discrete scheme is discussed for a Kelvin-Voigt viscoelastic fluid flow model with nonzero forcing function, which is either independent of time or in L ∞ (L 2). After deriving some a priori bounds for the solution of a semidiscrete Galerkin finite element scheme, a second-order backward difference method is applied for temporal discretization. Then, a priori estimates in Dirichlet norm are derived, which are valid uniformly in time using a combination of discrete Gronwall’s lemma and Stolz-Cesaro’s classical result on sequences. Moreover, an existence of a discrete global attractor for the discrete problem is established. Further, optimal a priori error estimates are obtained, whose bounds may depend exponentially in time. Under uniqueness condition, these estimates are shown to be uniform in time. Finally, several numerical experiments are conducted to confirm our theoretical findings.
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Acknowledgments
The author would like to express his gratitude to anonymous referees for their constructive and valuable suggestions, which help to improve the paper. The author also gratefully acknowledges the financial support from IRCC project No. 13IRAWD007 of IIT Bombay during his visit to IIT Bombay in 2016.
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Pany, A.K. Fully discrete second-order backward difference method for Kelvin-Voigt fluid flow model. Numer Algor 78, 1061–1086 (2018). https://doi.org/10.1007/s11075-017-0413-y
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DOI: https://doi.org/10.1007/s11075-017-0413-y