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October  2015, 20(8): 2583-2609. doi: 10.3934/dcdsb.2015.20.2583

## Fully discrete finite element method based on second-order Crank-Nicolson/Adams-Bashforth scheme for the equations of motion of Oldroyd fluids of order one

 1 School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China 2 Center for Computational Geosciences, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049

Received  October 2014 Revised  April 2015 Published  August 2015

In this paper, we study a fully discrete finite element method with second order accuracy in time for the equations of motion arising in the Oldroyd model of viscoelastic fluids. This method is based on a finite element approximation for the space discretization and the Crank-Nicolson/Adams-Bashforth scheme for the time discretization. The integral term is discretized by the trapezoidal rule to match with the second order accuracy in time. It leads to a linear system with a constant matrix and thus greatly increases the computational efficiency. Taking the nonnegativity of the quadrature rule and the technique of variable substitution for the trapezoidal rule approximation, we prove that this fully discrete finite element method is almost unconditionally stable and convergent. Furthermore, by the negative norm technique, we derive the $H^1$ and $L^2$-optimal error estimates of the velocity and the pressure.
Citation: Yingwen Guo, Yinnian He. Fully discrete finite element method based on second-order Crank-Nicolson/Adams-Bashforth scheme for the equations of motion of Oldroyd fluids of order one. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2583-2609. doi: 10.3934/dcdsb.2015.20.2583
##### References:
 [1] R. A. Adams, Sobolev Spaces,, Academic Press, (1975). [2] Yu. Ya. Agranovich and P. E. Sobolevskiĭ, Investigation of a mathematical model of a viscoelastic fluid,, Dokl. Akad. Nauk Ukrain. SSR Ser. A, 86 (1989), 3. [3] A. Ait Ou Ammi and M. Marion, Nonlinear Galerkin methods and mixed finite elements: Two-grid algorithms for the Navier-Stokes equations,, Numer. Math., 68 (1994), 189. doi: 10.1007/s002110050056. [4] P. G. Ciarlet, The Finite Element Method for Elliptic Problems,, North-Holland, (1978). [5] V. Girault and P. A. Raviart, Finite Element Method for Navier-Stokes Equations: Theory and Algorithms,, Springer-Verlag, (1986). doi: 10.1007/978-3-642-61623-5. [6] D. Goswami and A. K. Pani, A priori error estimates for semidiscrete finite element approxi- mations to the equations of motion arising in Oldroyd fluids of order one,, Int. J. Numer. Anal. Model., 8 (2011), 324. [7] D. Goswami and A. K. Pani, Backward Euler method for the equations of motion arising in Oldroyd fluids of order one with nonsmooth initial data,, preprint, (). [8] D. Goswami, A two-level finite element method for viscoelastic fluid flow: Non-smooth initial data,, preprint, (). [9] Y. He and K. Li, Convergence and stability of finite element nonlinear Galerkin method for the Navier-Stokes equations,, Numer. Math., 79 (1998), 77. doi: 10.1007/s002110050332. [10] Y. He, Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations,, SIAM J. Numer. Anal., 41 (2003), 1263. doi: 10.1137/S0036142901385659. [11] Y. He and K. M. Liu, A multilevel finite element method in space-time for the Navier-Stokes problem,, Numer. Methods Partial Differential Equations, 21 (2005), 1052. doi: 10.1002/num.20077. [12] Y. He, Y. Lin, S. S. P. Shen, W. Sun and R. Tait, Finite element approximation for the viscoelastic fluid motion problem,, J. Comput. Appl. Math., 155 (2003), 201. doi: 10.1016/S0377-0427(02)00864-6. [13] Y. He, Y. Lin, S. S. P. Shen and R. Tait, On the convergence of viscoelastic fluid flows to a steady state,, Adv. Differential Equations, 7 (2002), 717. [14] Y. He and W. Sun, Stability and convergence of the Crank-Nicolson/Adams-Bashforth scheme for the time-dependent Navier-Stokes equations,, SIAM J. Numer. Anal., 45 (2007), 837. doi: 10.1137/050639910. [15] J. G. Heywood and R. Rannacher, Finite-element approximations of the nonstationary Navier-Stokes problem. Part I: Regularity of solutions and second-order spatial discretization,, SIAM J. Numer. Anal., 19 (1982), 275. doi: 10.1137/0719018. [16] J. G. Heywood and R. Rannacher, Finite-element approximations of the nonstationary Navier-Stokes problem. Part IV: Error estimates for second-order time discretization,, SIAM J. Numer. Anal., 27 (1990), 353. doi: 10.1137/0727022. [17] A. T. Hill and E. Süli, Approximation of the global attractor for the incompressible Navier-Stokes equations,, IMA J. Numer. Anal., 20 (2000), 633. doi: 10.1093/imanum/20.4.633. [18] D. D. Joseph, Fluid Dynamics of Viscoelastic Liquids,, Springer Verlag, (1990). doi: 10.1007/978-1-4612-4462-2. [19] N. A. Karzeeva, A. A. Kot.siolis and A. P. Oskolkov, On dynamical system generated by initial-boundary value problems for the equations of motion of linear viscoelastic fluids,, Boundary value problems of mathematical physics, 188 (1990), 59. [20] R. B. Kellogg and J. E. Osborn, A regularity result for the Stokes problem in a convex polygon,, J. Functional Anal., 21 (1976), 397. doi: 10.1016/0022-1236(76)90035-5. [21] A. A. Kotsiolis, A. P. Oskolkov and R. D. Shadiev, A priori estimate on the semiaxis $t\geq0$ for the solutions of the equations of motion of linear viscoelastic fluids with an infinite Dirichlet integral and their applications,, J. Soviet Math., 62 (1992), 2777. doi: 10.1007/BF01671001. [22] S. Larsson, The long-time behavior of finite-element approximations of solutions to semilinear parabolic problems,, SIAM J. Numer. Anal., 26 (1989), 348. doi: 10.1137/0726019. [23] W. McLean and V. Thomée, Numerical solution of an evolution equation with a positive-type memory term,, J. Austral. Math. Soc. Ser. B, 35 (1993), 23. doi: 10.1017/S0334270000007268. [24] A. P. Oskolkov, Initial-boundary value problems for the equations of motion of Kelvin-Voigt fluids and Oldroyd fluids,, Boundary value problems of mathematical physics, 179 (1988), 126. [25] A. P. Oskolkov and D. V. Emel'yanova, Some nonlocal problems for two-dimensional equations of motion of Oldroyd fluids,, (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov., 189 (1991), 101. doi: 10.1007/BF01097499. [26] A. K. Pani and J. Y. Yuan, Semidiscrete finite element Galerkin approximations to the equations of motion arising in the Oldroyd model,, IMA J. Numer. Anal., 25 (2005), 750. doi: 10.1093/imanum/dri016. [27] A. K. Pani, J. Y. Yuan and P. Damazio, On a linearized backward Euler method for the equations of motion arising in the Oldroyd fluids of order one,, SIAM J. Numer. Anal., 44 (2006), 804. doi: 10.1137/S0036142903428967. [28] J. Shen, Long time stability and convergence for fully discrete nonlinear Galerkin methods,, Appl. Anal., 38 (1990), 201. doi: 10.1080/00036819008839963. [29] Z. Si, W. Li and Y. Wang, A gauge-Uzawa finite element method for the time-dependent Viscoelastic Oldroyd flows,, J. Math. Anal. Appl., 425 (2015), 96. doi: 10.1016/j.jmaa.2014.12.020. [30] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, AMS Chelsea Publishing, (1984). [31] K. Wang, Y. He and Y. Shang, Fully discrete finite element method for the viscoelastic fluid motion equations,, Discrete Continuous Dynam. Systems-B, 13 (2010), 665. doi: 10.3934/dcdsb.2010.13.665. [32] K. Wang, Y. He and X. Feng, On error estimates of the fully discrete penalty method for the viscoelastic flow problem,, Int. J. Comput. Math, 88 (2011), 2199. doi: 10.1080/00207160.2010.534781. [33] K. Wang, Y. Lin and Y. He, Asymptotic analysis of the equations of motion for viscoealstic Oldroyd fluid,, Discrete Contin. Dyn. Syst, 32 (2012), 657. [34] K. Wang, Z. Si and Y. Yang, Stabilized finite element method for the viscoelastic Oldroyd fluid flows,, Numer. Algorithms, 60 (2012), 75. doi: 10.1007/s11075-011-9512-3. [35] K. Wang, Y. He and Y. Lin, Long time numerical stability and asymptotic analysis for the viscoelastic Oldroyd flows,, Discrete Continuous Dynam. Systems-B, 17 (2012), 1551. doi: 10.3934/dcdsb.2012.17.1551. [36] W. L. Wilkinson, Non-Newtonian Fluids: Fluid Mechanics, Mixing and Heat Transfer,, Pergamon Press, (1960).

show all references

##### References:
 [1] R. A. Adams, Sobolev Spaces,, Academic Press, (1975). [2] Yu. Ya. Agranovich and P. E. Sobolevskiĭ, Investigation of a mathematical model of a viscoelastic fluid,, Dokl. Akad. Nauk Ukrain. SSR Ser. A, 86 (1989), 3. [3] A. Ait Ou Ammi and M. Marion, Nonlinear Galerkin methods and mixed finite elements: Two-grid algorithms for the Navier-Stokes equations,, Numer. Math., 68 (1994), 189. doi: 10.1007/s002110050056. [4] P. G. Ciarlet, The Finite Element Method for Elliptic Problems,, North-Holland, (1978). [5] V. Girault and P. A. Raviart, Finite Element Method for Navier-Stokes Equations: Theory and Algorithms,, Springer-Verlag, (1986). doi: 10.1007/978-3-642-61623-5. [6] D. Goswami and A. K. Pani, A priori error estimates for semidiscrete finite element approxi- mations to the equations of motion arising in Oldroyd fluids of order one,, Int. J. Numer. Anal. Model., 8 (2011), 324. [7] D. Goswami and A. K. Pani, Backward Euler method for the equations of motion arising in Oldroyd fluids of order one with nonsmooth initial data,, preprint, (). [8] D. Goswami, A two-level finite element method for viscoelastic fluid flow: Non-smooth initial data,, preprint, (). [9] Y. He and K. Li, Convergence and stability of finite element nonlinear Galerkin method for the Navier-Stokes equations,, Numer. Math., 79 (1998), 77. doi: 10.1007/s002110050332. [10] Y. He, Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations,, SIAM J. Numer. Anal., 41 (2003), 1263. doi: 10.1137/S0036142901385659. [11] Y. He and K. M. Liu, A multilevel finite element method in space-time for the Navier-Stokes problem,, Numer. Methods Partial Differential Equations, 21 (2005), 1052. doi: 10.1002/num.20077. [12] Y. He, Y. Lin, S. S. P. Shen, W. Sun and R. Tait, Finite element approximation for the viscoelastic fluid motion problem,, J. Comput. Appl. Math., 155 (2003), 201. doi: 10.1016/S0377-0427(02)00864-6. [13] Y. He, Y. Lin, S. S. P. Shen and R. Tait, On the convergence of viscoelastic fluid flows to a steady state,, Adv. Differential Equations, 7 (2002), 717. [14] Y. He and W. Sun, Stability and convergence of the Crank-Nicolson/Adams-Bashforth scheme for the time-dependent Navier-Stokes equations,, SIAM J. Numer. Anal., 45 (2007), 837. doi: 10.1137/050639910. [15] J. G. Heywood and R. Rannacher, Finite-element approximations of the nonstationary Navier-Stokes problem. Part I: Regularity of solutions and second-order spatial discretization,, SIAM J. Numer. Anal., 19 (1982), 275. doi: 10.1137/0719018. [16] J. G. Heywood and R. Rannacher, Finite-element approximations of the nonstationary Navier-Stokes problem. Part IV: Error estimates for second-order time discretization,, SIAM J. Numer. Anal., 27 (1990), 353. doi: 10.1137/0727022. [17] A. T. Hill and E. Süli, Approximation of the global attractor for the incompressible Navier-Stokes equations,, IMA J. Numer. Anal., 20 (2000), 633. doi: 10.1093/imanum/20.4.633. [18] D. D. Joseph, Fluid Dynamics of Viscoelastic Liquids,, Springer Verlag, (1990). doi: 10.1007/978-1-4612-4462-2. [19] N. A. Karzeeva, A. A. Kot.siolis and A. P. Oskolkov, On dynamical system generated by initial-boundary value problems for the equations of motion of linear viscoelastic fluids,, Boundary value problems of mathematical physics, 188 (1990), 59. [20] R. B. Kellogg and J. E. Osborn, A regularity result for the Stokes problem in a convex polygon,, J. Functional Anal., 21 (1976), 397. doi: 10.1016/0022-1236(76)90035-5. [21] A. A. Kotsiolis, A. P. Oskolkov and R. D. Shadiev, A priori estimate on the semiaxis $t\geq0$ for the solutions of the equations of motion of linear viscoelastic fluids with an infinite Dirichlet integral and their applications,, J. Soviet Math., 62 (1992), 2777. doi: 10.1007/BF01671001. [22] S. Larsson, The long-time behavior of finite-element approximations of solutions to semilinear parabolic problems,, SIAM J. Numer. Anal., 26 (1989), 348. doi: 10.1137/0726019. [23] W. McLean and V. Thomée, Numerical solution of an evolution equation with a positive-type memory term,, J. Austral. Math. Soc. Ser. B, 35 (1993), 23. doi: 10.1017/S0334270000007268. [24] A. P. Oskolkov, Initial-boundary value problems for the equations of motion of Kelvin-Voigt fluids and Oldroyd fluids,, Boundary value problems of mathematical physics, 179 (1988), 126. [25] A. P. Oskolkov and D. V. Emel'yanova, Some nonlocal problems for two-dimensional equations of motion of Oldroyd fluids,, (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov., 189 (1991), 101. doi: 10.1007/BF01097499. [26] A. K. Pani and J. Y. Yuan, Semidiscrete finite element Galerkin approximations to the equations of motion arising in the Oldroyd model,, IMA J. Numer. Anal., 25 (2005), 750. doi: 10.1093/imanum/dri016. [27] A. K. Pani, J. Y. Yuan and P. Damazio, On a linearized backward Euler method for the equations of motion arising in the Oldroyd fluids of order one,, SIAM J. Numer. Anal., 44 (2006), 804. doi: 10.1137/S0036142903428967. [28] J. Shen, Long time stability and convergence for fully discrete nonlinear Galerkin methods,, Appl. Anal., 38 (1990), 201. doi: 10.1080/00036819008839963. [29] Z. Si, W. Li and Y. Wang, A gauge-Uzawa finite element method for the time-dependent Viscoelastic Oldroyd flows,, J. Math. Anal. Appl., 425 (2015), 96. doi: 10.1016/j.jmaa.2014.12.020. [30] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, AMS Chelsea Publishing, (1984). [31] K. Wang, Y. He and Y. Shang, Fully discrete finite element method for the viscoelastic fluid motion equations,, Discrete Continuous Dynam. Systems-B, 13 (2010), 665. doi: 10.3934/dcdsb.2010.13.665. [32] K. Wang, Y. He and X. Feng, On error estimates of the fully discrete penalty method for the viscoelastic flow problem,, Int. J. Comput. Math, 88 (2011), 2199. doi: 10.1080/00207160.2010.534781. [33] K. Wang, Y. Lin and Y. He, Asymptotic analysis of the equations of motion for viscoealstic Oldroyd fluid,, Discrete Contin. Dyn. Syst, 32 (2012), 657. [34] K. Wang, Z. Si and Y. Yang, Stabilized finite element method for the viscoelastic Oldroyd fluid flows,, Numer. Algorithms, 60 (2012), 75. doi: 10.1007/s11075-011-9512-3. [35] K. Wang, Y. He and Y. Lin, Long time numerical stability and asymptotic analysis for the viscoelastic Oldroyd flows,, Discrete Continuous Dynam. Systems-B, 17 (2012), 1551. doi: 10.3934/dcdsb.2012.17.1551. [36] W. L. Wilkinson, Non-Newtonian Fluids: Fluid Mechanics, Mixing and Heat Transfer,, Pergamon Press, (1960).
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