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Fully discrete second-order backward difference method for Kelvin-Voigt fluid flow model

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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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References

  1. Abboud, H., Girault, V., Sayah, T.: A second order accuracy in time for a full discretized time-dependent Navier-Stokes equations by a two-grid scheme. Numer. Math. 114, 189–231 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Akhmatov, M.M., Oskolkov, A.P.: On convergence difference schemes for the equations of motion of an Oldroyd fluid. J. Math. Sci. (New York) 47, 2926–2933 (1989)

    MATH  Google Scholar 

  3. Bajpai, S., Nataraj, N., Pani, A.K., Damazio, P., Yuan, J.Y.: Semidiscrete Galerkin method for equations of motion arising in Kelvin-Voigt model of viscoelastic fluid flow. Numer. Meth. PDEs. 29, 857–883 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bajpai, S., Nataraj, N., Pani, A.K.: On fully discrete finite element schemes for equation of motion of Kelvin-Voight fluids. Intnl. J. Numer. Anal. Mod. 10, 481–507 (2013)

    MATH  Google Scholar 

  5. Bercovier, M., Pironneau, O.: Error estimates for finite element solution of the Stokes problem in the primitive variables. Numer. Math. 33, 211–224 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)

    Book  MATH  Google Scholar 

  7. Burtscher, M., Szczyrba, I.: Numerical modeling of brain dynamics in traumatic situations - impulsive translations. In: The 2005 International Conference on Mathematics and Engineering Techniques in Medicine and Biological Sciences, pp 205–211 (2005)

  8. Burtscher, M., Szczyrba, I.: Computational simulation and visualization of traumatic brain injuries. In: International Conference on Modeling, Simulation and Visualization Methods, pp 101–107 (2006)

  9. Cotter, C.S., Smolarkiewicz, P.K., Szezyrba, I.N.: A viscoelastic model from brain injuries. Internat. J. Numer. Methods Fluids 40, 303–311 (2002)

    Article  MATH  Google Scholar 

  10. Girault, V., Raviart, P.A.: Finite Element Approximation of the Navier-Stokes Equations. Lecture Notes in Mathematics, No. 749. Springer, New York (1980)

    Google Scholar 

  11. Goswami, D., Pani, A.K.: A priori error estimates for semidiscrete finite element approximation to equations of motion arising in Oldroyd fluids of order one. Intnl. J. Numer. Anal. Mod. 8, 324–352 (2011)

    MathSciNet  MATH  Google Scholar 

  12. Guo, Y., He, Y.: Fully discrete finite element method based on second order Crank-Nicolson/Adams-Bashforth scheme for the equations of motion of Oldroyd fluid of order one. Discrete. Contin. Dyn. Sys. Ser. B 20, 2583–2609 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. He, Y.: The Euler implicit/explicit scheme for 2D time dependent Navier-Stokes equations with smooth or non-smooth initial data. Math. Comp. 77, 2097–2124 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. He, Y., Lin, Y., Shen, S.S.P., Sun, W., Tait, R.: Finite element approximation for the viscoelastic fluid motion problem. J. Comput. Appl. Math. 155, 201–222 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem: I. Regularity of solutions and second order error estimates for spatial discretization. SIAM J. Numer. Anal. 19, 275–311 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  16. Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem: IV: Error analysis for second-order time discretization. SIAM J. Numer. Anal. 27, 353–384 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kalantarov, V.K., Titi, E.S.: Global attractors and determining modes for the 3D Navier-Stokes-Voight equations. Chinese Annals of Mathematics Series B 30, 697–714 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kesavan, S.: Topics in functional analysis and applications. Wiley (1989)

  19. Ladyzenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York (1969)

    Google Scholar 

  20. Oskolkov, A.P.: Theory of nonstationary flows of Kelvin-Voigt fluids. J. Math. Sciences 28, 751–758 (1985)

    MATH  Google Scholar 

  21. Oskolkov, A.P.: Initial-boundary value problems for equations of motion of Kelvin–Voigt fluids and Oldroyd fluids. Tr. Mat. Inst. Steklova 179, 126–164 (1987)

    Google Scholar 

  22. Oskolkov, A.P.: On an estimate, uniform on the semiaxis t ≥ 0, for the rate of convergence of Galerkin approximations for the equations of motion of Kelvin-Voight fluids. J. Math. Sciences 62, 2802–2806 (1992)

    MathSciNet  Google Scholar 

  23. Oskolkov, A.P., Shadiev, R.D.: Non local problems in the theory of the motion equations of Kelvin-Voight fluids. J. Math. Sciences 59, 1206–1214 (1992)

    Google Scholar 

  24. Oskolkov, A.P., Shadiev, R.D.: Towards a theory of global solvability on [0,∞] of initial-boundary value problems for the equations of motion of Oldroyd and Kelvin-Voight fluids. J. Math. Sci. Univ. Tokyo 68, 240–253 (1994)

    Article  MATH  Google Scholar 

  25. Pani, A.K., Yuan, J.Y.: Semidiscrete finite element Galerkin approximation to the equations of motion arising in the Oldroyd model. IMA J. Numer. Anal. 25, 750–782 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pani, A.K., Yuan, J.Y., Damazio, P.D.: On a linearized backward Euler method for the equations of motion of oldroyd fluids of order one. SIAM J. Numer. Anal. 44, 804–825 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Pani, A.K., Pany, A.K., Damazio, P., Yuan, J.Y.: A modified nonlinear spectral Galerkin method for the equations of motion arising in the Kelvin-Voigt fluids. Applicable Anal. 93, 1587–1610 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Pany, A.K., Bajpai, S., Pani, A.K.: Optimal error estimates for the semidiscrete Galerkin approximations to the equations of motion described by Kelvin-Voigt viscoelastic fluid flow model. J. Comp. Appl. Math. 302, 234–257 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  29. Pany, A.K., Paikray, S., Padhy, S., Pani, A.K.: Backward Euler schemes for the Kelvin-Voigt viscoelastic fluid flow model. Intnl. J. Numer. Anal. Mod. 14, 126–151 (2017)

    MathSciNet  MATH  Google Scholar 

  30. Temam, R.: Navier-Stokes Equations, Theory and Numerical Analysis. North-Holland, Amsterdam (1984)

    MATH  Google Scholar 

  31. Wang, K., He, Y., Shang, Y.: Fully discrete finite element method for the viscoelastic fluid motion equations. Discrete. Contin. Dyn. Sys. Ser. B 13, 665–684 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, K., He, Y., Feng, X.: On error estimates of the penalty method for the viscoelastic flow problem I: time discretization. Appl. Math. Model. 34, 4089–4105 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wang, K., He, Y., Feng, X.: On error estimates of the fully discrete penalty method for the viscoelastic flow problem. Int. J. Comput. Math 88, 2199–2220 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wang, K., Lin, Y., He, Y.: Asymptotic analysis of the equations of motion for viscoelastic oldroyd fluid. Discrete Contin. Dyn. Syst. 32, 657–677 (2012)

    MathSciNet  MATH  Google Scholar 

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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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  1. Center for Applied Mathematics, Siksha O Anusandhan University, Bhubaneswar, Odisha, 751030, India

    Ambit Kumar Pany

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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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