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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).   Google Scholar [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.   Google Scholar [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.  Google Scholar [4] P. G. Ciarlet, The Finite Element Method for Elliptic Problems,, North-Holland, (1978).   Google Scholar [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.  Google Scholar [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.   Google Scholar [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, ().   Google Scholar [8] D. Goswami, A two-level finite element method for viscoelastic fluid flow: Non-smooth initial data,, preprint, ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] D. D. Joseph, Fluid Dynamics of Viscoelastic Liquids,, Springer Verlag, (1990).  doi: 10.1007/978-1-4612-4462-2.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] J. Shen, Long time stability and convergence for fully discrete nonlinear Galerkin methods,, Appl. Anal., 38 (1990), 201.  doi: 10.1080/00036819008839963.  Google Scholar [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.  Google Scholar [30] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, AMS Chelsea Publishing, (1984).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [36] W. L. Wilkinson, Non-Newtonian Fluids: Fluid Mechanics, Mixing and Heat Transfer,, Pergamon Press, (1960).   Google Scholar

show all references

##### References:
 [1] R. A. Adams, Sobolev Spaces,, Academic Press, (1975).   Google Scholar [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.   Google Scholar [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.  Google Scholar [4] P. G. Ciarlet, The Finite Element Method for Elliptic Problems,, North-Holland, (1978).   Google Scholar [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.  Google Scholar [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.   Google Scholar [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, ().   Google Scholar [8] D. Goswami, A two-level finite element method for viscoelastic fluid flow: Non-smooth initial data,, preprint, ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] D. D. Joseph, Fluid Dynamics of Viscoelastic Liquids,, Springer Verlag, (1990).  doi: 10.1007/978-1-4612-4462-2.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] J. Shen, Long time stability and convergence for fully discrete nonlinear Galerkin methods,, Appl. Anal., 38 (1990), 201.  doi: 10.1080/00036819008839963.  Google Scholar [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.  Google Scholar [30] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, AMS Chelsea Publishing, (1984).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [36] W. L. Wilkinson, Non-Newtonian Fluids: Fluid Mechanics, Mixing and Heat Transfer,, Pergamon Press, (1960).   Google Scholar
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