# American Institute of Mathematical Sciences

July  2012, 17(5): 1551-1573. doi: 10.3934/dcdsb.2012.17.1551

## Long time numerical stability and asymptotic analysis for the viscoelastic Oldroyd flows

 1 College of Mathematics and Statistics, Chongqing University, Chongqing 401331 2 Center for Computational Geosciences, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China 3 Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, China

Received  August 2010 Revised  December 2011 Published  March 2012

In this article, we study the long time numerical stability and asymptotic behavior for the viscoelastic Oldroyd fluid motion equations. Firstly, with the Euler semi-implicit scheme for the temporal discretization, we deduce the global $H^2-$stability result for the fully discrete finite element solution. Secondly, based on the uniform stability of the numerical solution, we investigate the discrete asymptotic behavior and claim that the viscoelastic Oldroyd problem converges to the stationary Navier-Stokes flows if the body force $f(x,t)$ approaches to a steady-state $f_\infty(x)$ as $t\rightarrow\infty$. Finally, some numerical experiments are given to verify the theoretical predictions.
Citation: Kun Wang, Yinnian He, Yanping Lin. Long time numerical stability and asymptotic analysis for the viscoelastic Oldroyd flows. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1551-1573. doi: 10.3934/dcdsb.2012.17.1551
##### References:
 [1] Y. Agranovich and P. Sobolevskii, Motion of non-linear viscoelastic fluid, Nonlinear Anal., 32 (1998), 755-760. doi: 10.1016/S0362-546X(97)00519-1. [2] M. Akhmatov and A. Oskolkov, On convergence difference schemes for the equations of motion of an Oldroyd fluid, LOMI, 159 (1987), 143-152; translation in J. Soviet. Math., 47 (1989), 2926-2933. doi: 10.1007/BF01305224. [3] W. Allegretto, Y. Lin and A. Zhou, Long-time stability of finite element approximations for parabolic equations with memory, Numer. Methods Partial Differential Eq., 15 (1999), 333-354. [4] G. Araújo, S. de Menezes and A. Marinho, Existence of solutions for an Oldroyd model of viscoelastic fluids,, Electron J. Differential Equations, 2009 (). [5] R. Bird, R. Armstrong and O. Hassager, "Dynamics of Polymeric Liquids. Vol. 1. Fluid Mechanics," John Wiley & sons, New York, 1977. [6] J. Cannon, R. Ewing, Y. He and Y. Lin, A modified nonlinear Galerkin method for the viscoelastic fluid motion equations, Int. J. Eng. Sci., 37 (1999), 1643-1662. doi: 10.1016/S0020-7225(98)00142-6. [7] P. Ciarlet, "The Finite Element Method for Elliptic Problems," Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. [8] V. Girault and P.-A. Raviart, "Finite Element Approximation of the Navier-Stokes Equations. Theory and Algorithms," Springer Series in Computational Mathematics, 5, Springer-Verlag, Berlin, 1979. [9] D. Goswami and A. Pani, A priori error estimates for semidiscrete finite element approximations to equations of motion arising in Oldroyd fluids of order one, Int. J. Numer. Anal. Model., 8 (2011), 324-352. [10] Y. He and J. Li, Convergence of three iterative methods based on the finite element discretization for the stationary Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 198 (2009), 1351-1359. [11] Y. He and J. Li, Two-level methods based on three corrections for the 2D/3D steady Navier-Stokes equations, Int. J. Numer. Anal. Model. Ser. B, 2 (2011), 42-56. [12] Y. He and Y. Li, Asymptotic behavior of linearized viscoelastic flow problem, Discrete Contin. Dyn. Syst.-Ser. B, 10 (2008), 843-856. [13] Y. He and K. Li, Asymptotic behavior and time discretization analysis for the non-stationary Navier-Stokes problem, Numer. Math., 98 (2004), 647-673. doi: 10.1007/s00211-004-0532-y. [14] Y. He, Y. Lin, S. Shen and R. Tait, On the convergence of viscoelastic fluid flows to a steady state, Adv. Differential Equations, 7 (2002), 717-742. [15] Y. He, Y. Lin, S. Shen, W. Sun and R. Tait, Finite element approximation for the viscoelastic fluid motion problem, J. Comput. Appl. Math., 155 (2003), 201-222. doi: 10.1016/S0377-0427(02)00864-6. [16] 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), 837-869. doi: 10.1137/050639910. [17] J. Heywood and R. Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem part IV: error analysis for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), 353-384. doi: 10.1137/0727022. [18] D. Joseph, "Fluid Dynamics of Viscoelastic Liquids," Springer-Verlag, New York, 1990. [19] A. Kotsiolis and A. Oskolkov, Initial boundary-value problems for equations of slightly compressible Jeffreys-Oldroyd fluids, Zap. Nauchn. Semin. POMI 208 (1993) 200-218, J. Math. Sci., 81 (1996), 2578-2588. doi: 10.1007/BF02362429. [20] N. Ju, On the global stability of a temporal discretization scheme for the Navier-Stokes equations, IMA J. Numer. Anal., 22 (2002), 577-597. doi: 10.1093/imanum/22.4.577. [21] J. Li, J. Wu, Z. Chen and A. Wang, Superconvergence of stabilized low order finite volume approximation for the three-dimensional stationary Navier-Stokes equations, Int. J. Numer. Anal. Model., 9 (2012), 419-431. [22] Y. Li and K. Li, Operator splitting methods for the Navier-Stokes equations with nonlinear slip boundary conditions, Int. J. Numer. Anal. Model., 7 (2010), 785-805. [23] J. Oldroyd, On the formulation of the rheological equations of state, Proc. Roy. Soc. Lond. Math. Phys. Sci., 200 (1950), 523-541. doi: 10.1098/rspa.1950.0035. [24] A. Oskolkov, Initial boundary value problems for the equations of motion of Kelvin-Voigt fluids and Oldroyd fluids, Contributions to "Boundary Value Problems of Mathematical Physics"(ed. J. Schulenberger), Proc. Steklov Inst. Math., 2 (1989), 137-182. [25] A. Oskolkov, The penalty method for equations of viscoelastic media, Zap. Nauchn. Semin. POMI 224 (1995) 267-278, J. Math. Sci., 88 (1998), 283-291. doi: 10.1007/BF02364990. [26] A. Pani and J. Yuan, Semidiscrete finite element Galerkin approximation to the equations of motion arising in the Oldroyd model, IMA J. Numer. Anal., 25 (2005), 750-782. doi: 10.1093/imanum/dri016. [27] A. Pani, J. Yuan and P. Damazio, On a linearized backward Euler method for the equations of motion of Oldroyd fluids of order one, SIAM J. Numer. Anal., 44 (2006), 804-825. doi: 10.1137/S0036142903428967. [28] L. Shen, J. Li and Z. Chen, Analysis of a stabilized finite volume method for the transient Stokes equations, Int. J. Numer. Anal. Model., 6 (2009), 505-519. [29] J. Shen, Long time stability and convergence for fully discrete nonlinear Galerkin methods, Appl. Anal., 38 (1990), 201-229. doi: 10.1080/00036819008839963. [30] P. Sobolevskii, Stabilization of viscoelastic fluid motion (Oldroyd's mathematical model), Differential Integral Equations, 7 (1994), 1597-1612. [31] P. Sobolevskii, Asymptotic of stable viscoelastic fluid motion (Oldroyd's mathematical model), Math. Nachr., 177 (1996), 281-305. doi: 10.1002/mana.19961770116. [32] H. Sun, Y. He and X. Feng, On error estimates of the pressure-correction projection methods for the time-dependent Navier-Stokes equations, Int. J. Numer. Anal. Model., 8 (2011), 70-85. [33] R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," North-Holland, Amsterdam, 1984. [34] R. Temam and X. Wang, Boundary layers asscociated with incompressible Navier-Stokes equations: the noncharacteristic boundary case, J. Differential Equations, 179 (2002), 647-686. doi: 10.1006/jdeq.2001.4038. [35] F. Tone and D. Wirosoetisno, On the long-time staility of the implicit Euler scheme for the two-dimensional Navier-Stokes equations, SIAM J. Numer. Anal., 44 (2006), 29-40. doi: 10.1137/040618527. [36] K. Wang, Y. He and X. Feng, On error estimates of the penalty method for the viscoelastic flow problem I: time discretization, Appl. Math. Modell., 34 (2010), 4089-4105. doi: 10.1016/j.apm.2010.04.008. [37] K. Wang, Y. He and X. Feng, On error estimates of the fully discrete penalty method for the viscoelastic flow problem, Internat. J. Comput. Math., 88 (2011), 2199-2220. doi: 10.1080/00207160.2010.534781. [38] K. Wang, Y. He and Y. Shang, Fully discrete finite element method for the viscoelastic fluid motion equations, Discrete Contin. Dyn. Syst.-Ser. B, 13 (2010), 665-684. [39] K. Wang, Y. Lin and Y. He, Asymptotic analysis of the equations of motion for viscoelastic Oldroyd fluid, Discete Contin. Dyn. Syst., 32 (2012), 657-677. [40] K. Wang, Z. Si and Y. Yang, Stabilized finite element method for the viscoelastic Oldroyd fluid flows, Numer. Algor., (2011). doi: 10.1007/s11075-011-9512-3.

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##### References:
 [1] Y. Agranovich and P. Sobolevskii, Motion of non-linear viscoelastic fluid, Nonlinear Anal., 32 (1998), 755-760. doi: 10.1016/S0362-546X(97)00519-1. [2] M. Akhmatov and A. Oskolkov, On convergence difference schemes for the equations of motion of an Oldroyd fluid, LOMI, 159 (1987), 143-152; translation in J. Soviet. Math., 47 (1989), 2926-2933. doi: 10.1007/BF01305224. [3] W. Allegretto, Y. Lin and A. Zhou, Long-time stability of finite element approximations for parabolic equations with memory, Numer. Methods Partial Differential Eq., 15 (1999), 333-354. [4] G. Araújo, S. de Menezes and A. Marinho, Existence of solutions for an Oldroyd model of viscoelastic fluids,, Electron J. Differential Equations, 2009 (). [5] R. Bird, R. Armstrong and O. Hassager, "Dynamics of Polymeric Liquids. Vol. 1. Fluid Mechanics," John Wiley & sons, New York, 1977. [6] J. Cannon, R. Ewing, Y. He and Y. Lin, A modified nonlinear Galerkin method for the viscoelastic fluid motion equations, Int. J. Eng. Sci., 37 (1999), 1643-1662. doi: 10.1016/S0020-7225(98)00142-6. [7] P. Ciarlet, "The Finite Element Method for Elliptic Problems," Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. [8] V. Girault and P.-A. Raviart, "Finite Element Approximation of the Navier-Stokes Equations. Theory and Algorithms," Springer Series in Computational Mathematics, 5, Springer-Verlag, Berlin, 1979. [9] D. Goswami and A. Pani, A priori error estimates for semidiscrete finite element approximations to equations of motion arising in Oldroyd fluids of order one, Int. J. Numer. Anal. Model., 8 (2011), 324-352. [10] Y. He and J. Li, Convergence of three iterative methods based on the finite element discretization for the stationary Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 198 (2009), 1351-1359. [11] Y. He and J. Li, Two-level methods based on three corrections for the 2D/3D steady Navier-Stokes equations, Int. J. Numer. Anal. Model. Ser. B, 2 (2011), 42-56. [12] Y. He and Y. Li, Asymptotic behavior of linearized viscoelastic flow problem, Discrete Contin. Dyn. Syst.-Ser. B, 10 (2008), 843-856. [13] Y. He and K. Li, Asymptotic behavior and time discretization analysis for the non-stationary Navier-Stokes problem, Numer. Math., 98 (2004), 647-673. doi: 10.1007/s00211-004-0532-y. [14] Y. He, Y. Lin, S. Shen and R. Tait, On the convergence of viscoelastic fluid flows to a steady state, Adv. Differential Equations, 7 (2002), 717-742. [15] Y. He, Y. Lin, S. Shen, W. Sun and R. Tait, Finite element approximation for the viscoelastic fluid motion problem, J. Comput. Appl. Math., 155 (2003), 201-222. doi: 10.1016/S0377-0427(02)00864-6. [16] 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), 837-869. doi: 10.1137/050639910. [17] J. Heywood and R. Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem part IV: error analysis for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), 353-384. doi: 10.1137/0727022. [18] D. Joseph, "Fluid Dynamics of Viscoelastic Liquids," Springer-Verlag, New York, 1990. [19] A. Kotsiolis and A. Oskolkov, Initial boundary-value problems for equations of slightly compressible Jeffreys-Oldroyd fluids, Zap. Nauchn. Semin. POMI 208 (1993) 200-218, J. Math. Sci., 81 (1996), 2578-2588. doi: 10.1007/BF02362429. [20] N. Ju, On the global stability of a temporal discretization scheme for the Navier-Stokes equations, IMA J. Numer. Anal., 22 (2002), 577-597. doi: 10.1093/imanum/22.4.577. [21] J. Li, J. Wu, Z. Chen and A. Wang, Superconvergence of stabilized low order finite volume approximation for the three-dimensional stationary Navier-Stokes equations, Int. J. Numer. Anal. Model., 9 (2012), 419-431. [22] Y. Li and K. Li, Operator splitting methods for the Navier-Stokes equations with nonlinear slip boundary conditions, Int. J. Numer. Anal. Model., 7 (2010), 785-805. [23] J. Oldroyd, On the formulation of the rheological equations of state, Proc. Roy. Soc. Lond. Math. Phys. Sci., 200 (1950), 523-541. doi: 10.1098/rspa.1950.0035. [24] A. Oskolkov, Initial boundary value problems for the equations of motion of Kelvin-Voigt fluids and Oldroyd fluids, Contributions to "Boundary Value Problems of Mathematical Physics"(ed. J. Schulenberger), Proc. Steklov Inst. Math., 2 (1989), 137-182. [25] A. Oskolkov, The penalty method for equations of viscoelastic media, Zap. Nauchn. Semin. POMI 224 (1995) 267-278, J. Math. Sci., 88 (1998), 283-291. doi: 10.1007/BF02364990. [26] A. Pani and J. Yuan, Semidiscrete finite element Galerkin approximation to the equations of motion arising in the Oldroyd model, IMA J. Numer. Anal., 25 (2005), 750-782. doi: 10.1093/imanum/dri016. [27] A. Pani, J. Yuan and P. Damazio, On a linearized backward Euler method for the equations of motion of Oldroyd fluids of order one, SIAM J. Numer. Anal., 44 (2006), 804-825. doi: 10.1137/S0036142903428967. [28] L. Shen, J. Li and Z. Chen, Analysis of a stabilized finite volume method for the transient Stokes equations, Int. J. Numer. Anal. Model., 6 (2009), 505-519. [29] J. Shen, Long time stability and convergence for fully discrete nonlinear Galerkin methods, Appl. Anal., 38 (1990), 201-229. doi: 10.1080/00036819008839963. [30] P. Sobolevskii, Stabilization of viscoelastic fluid motion (Oldroyd's mathematical model), Differential Integral Equations, 7 (1994), 1597-1612. [31] P. Sobolevskii, Asymptotic of stable viscoelastic fluid motion (Oldroyd's mathematical model), Math. Nachr., 177 (1996), 281-305. doi: 10.1002/mana.19961770116. [32] H. Sun, Y. He and X. Feng, On error estimates of the pressure-correction projection methods for the time-dependent Navier-Stokes equations, Int. J. Numer. Anal. Model., 8 (2011), 70-85. [33] R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," North-Holland, Amsterdam, 1984. [34] R. Temam and X. Wang, Boundary layers asscociated with incompressible Navier-Stokes equations: the noncharacteristic boundary case, J. Differential Equations, 179 (2002), 647-686. doi: 10.1006/jdeq.2001.4038. [35] F. Tone and D. Wirosoetisno, On the long-time staility of the implicit Euler scheme for the two-dimensional Navier-Stokes equations, SIAM J. Numer. Anal., 44 (2006), 29-40. doi: 10.1137/040618527. [36] K. Wang, Y. He and X. Feng, On error estimates of the penalty method for the viscoelastic flow problem I: time discretization, Appl. Math. Modell., 34 (2010), 4089-4105. doi: 10.1016/j.apm.2010.04.008. [37] K. Wang, Y. He and X. Feng, On error estimates of the fully discrete penalty method for the viscoelastic flow problem, Internat. J. Comput. Math., 88 (2011), 2199-2220. doi: 10.1080/00207160.2010.534781. [38] K. Wang, Y. He and Y. Shang, Fully discrete finite element method for the viscoelastic fluid motion equations, Discrete Contin. Dyn. Syst.-Ser. B, 13 (2010), 665-684. [39] K. Wang, Y. Lin and Y. He, Asymptotic analysis of the equations of motion for viscoelastic Oldroyd fluid, Discete Contin. Dyn. Syst., 32 (2012), 657-677. [40] K. Wang, Z. Si and Y. Yang, Stabilized finite element method for the viscoelastic Oldroyd fluid flows, Numer. Algor., (2011). doi: 10.1007/s11075-011-9512-3.
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