# 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 & 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. doi: 10.1016/S0362-546X(97)00519-1. Google Scholar [2] M. Akhmatov and A. Oskolkov, On convergence difference schemes for the equations of motion of an Oldroyd fluid,, LOMI, 159 (1987), 143. doi: 10.1007/BF01305224. Google Scholar [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. Google Scholar [4] G. Araújo, S. de Menezes and A. Marinho, Existence of solutions for an Oldroyd model of viscoelastic fluids,, Electron J. Differential Equations, 2009 (). Google Scholar [5] R. Bird, R. Armstrong and O. Hassager, "Dynamics of Polymeric Liquids. Vol. 1. Fluid Mechanics,", John Wiley & sons, (1977). Google Scholar [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. doi: 10.1016/S0020-7225(98)00142-6. Google Scholar [7] P. Ciarlet, "The Finite Element Method for Elliptic Problems,", Studies in Mathematics and its Applications, (1978). Google Scholar [8] V. Girault and P.-A. Raviart, "Finite Element Approximation of the Navier-Stokes Equations. Theory and Algorithms,", Springer Series in Computational Mathematics, 5 (1979). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] Y. He and Y. Li, Asymptotic behavior of linearized viscoelastic flow problem,, Discrete Contin. Dyn. Syst.-Ser. B, 10 (2008), 843. Google Scholar [13] Y. He and K. Li, Asymptotic behavior and time discretization analysis for the non-stationary Navier-Stokes problem,, Numer. Math., 98 (2004), 647. doi: 10.1007/s00211-004-0532-y. Google Scholar [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. Google Scholar [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. doi: 10.1016/S0377-0427(02)00864-6. Google Scholar [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. doi: 10.1137/050639910. Google Scholar [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. doi: 10.1137/0727022. Google Scholar [18] D. Joseph, "Fluid Dynamics of Viscoelastic Liquids,", Springer-Verlag, (1990). Google Scholar [19] A. Kotsiolis and A. Oskolkov, Initial boundary-value problems for equations of slightly compressible Jeffreys-Oldroyd fluids,, Zap. Nauchn. Semin. POMI \textbf{208} (1993) 200-218, 208 (1993), 200. doi: 10.1007/BF02362429. Google Scholar [20] N. Ju, On the global stability of a temporal discretization scheme for the Navier-Stokes equations,, IMA J. Numer. Anal., 22 (2002), 577. doi: 10.1093/imanum/22.4.577. Google Scholar [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. Google Scholar [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. Google Scholar [23] J. Oldroyd, On the formulation of the rheological equations of state,, Proc. Roy. Soc. Lond. Math. Phys. Sci., 200 (1950), 523. doi: 10.1098/rspa.1950.0035. Google Scholar [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. Google Scholar [25] A. Oskolkov, The penalty method for equations of viscoelastic media,, Zap. Nauchn. Semin. POMI \textbf{224} (1995) 267-278, 224 (1995), 267. doi: 10.1007/BF02364990. Google Scholar [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. doi: 10.1093/imanum/dri016. Google Scholar [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. doi: 10.1137/S0036142903428967. Google Scholar [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. Google Scholar [29] J. Shen, Long time stability and convergence for fully discrete nonlinear Galerkin methods,, Appl. Anal., 38 (1990), 201. doi: 10.1080/00036819008839963. Google Scholar [30] P. Sobolevskii, Stabilization of viscoelastic fluid motion (Oldroyd's mathematical model),, Differential Integral Equations, 7 (1994), 1597. Google Scholar [31] P. Sobolevskii, Asymptotic of stable viscoelastic fluid motion (Oldroyd's mathematical model),, Math. Nachr., 177 (1996), 281. doi: 10.1002/mana.19961770116. Google Scholar [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. Google Scholar [33] R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", North-Holland, (1984). Google Scholar [34] R. Temam and X. Wang, Boundary layers asscociated with incompressible Navier-Stokes equations: the noncharacteristic boundary case,, J. Differential Equations, 179 (2002), 647. doi: 10.1006/jdeq.2001.4038. Google Scholar [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. doi: 10.1137/040618527. Google Scholar [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. doi: 10.1016/j.apm.2010.04.008. Google Scholar [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. doi: 10.1080/00207160.2010.534781. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

show all references

##### References:
 [1] Y. Agranovich and P. Sobolevskii, Motion of non-linear viscoelastic fluid,, Nonlinear Anal., 32 (1998), 755. doi: 10.1016/S0362-546X(97)00519-1. Google Scholar [2] M. Akhmatov and A. Oskolkov, On convergence difference schemes for the equations of motion of an Oldroyd fluid,, LOMI, 159 (1987), 143. doi: 10.1007/BF01305224. Google Scholar [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. Google Scholar [4] G. Araújo, S. de Menezes and A. Marinho, Existence of solutions for an Oldroyd model of viscoelastic fluids,, Electron J. Differential Equations, 2009 (). Google Scholar [5] R. Bird, R. Armstrong and O. Hassager, "Dynamics of Polymeric Liquids. Vol. 1. Fluid Mechanics,", John Wiley & sons, (1977). Google Scholar [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. doi: 10.1016/S0020-7225(98)00142-6. Google Scholar [7] P. Ciarlet, "The Finite Element Method for Elliptic Problems,", Studies in Mathematics and its Applications, (1978). Google Scholar [8] V. Girault and P.-A. Raviart, "Finite Element Approximation of the Navier-Stokes Equations. Theory and Algorithms,", Springer Series in Computational Mathematics, 5 (1979). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] Y. He and Y. Li, Asymptotic behavior of linearized viscoelastic flow problem,, Discrete Contin. Dyn. Syst.-Ser. B, 10 (2008), 843. Google Scholar [13] Y. He and K. Li, Asymptotic behavior and time discretization analysis for the non-stationary Navier-Stokes problem,, Numer. Math., 98 (2004), 647. doi: 10.1007/s00211-004-0532-y. Google Scholar [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. Google Scholar [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. doi: 10.1016/S0377-0427(02)00864-6. Google Scholar [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. doi: 10.1137/050639910. Google Scholar [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. doi: 10.1137/0727022. Google Scholar [18] D. Joseph, "Fluid Dynamics of Viscoelastic Liquids,", Springer-Verlag, (1990). Google Scholar [19] A. Kotsiolis and A. Oskolkov, Initial boundary-value problems for equations of slightly compressible Jeffreys-Oldroyd fluids,, Zap. Nauchn. Semin. POMI \textbf{208} (1993) 200-218, 208 (1993), 200. doi: 10.1007/BF02362429. Google Scholar [20] N. Ju, On the global stability of a temporal discretization scheme for the Navier-Stokes equations,, IMA J. Numer. Anal., 22 (2002), 577. doi: 10.1093/imanum/22.4.577. Google Scholar [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. Google Scholar [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. Google Scholar [23] J. Oldroyd, On the formulation of the rheological equations of state,, Proc. Roy. Soc. Lond. Math. Phys. Sci., 200 (1950), 523. doi: 10.1098/rspa.1950.0035. Google Scholar [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. Google Scholar [25] A. Oskolkov, The penalty method for equations of viscoelastic media,, Zap. Nauchn. Semin. POMI \textbf{224} (1995) 267-278, 224 (1995), 267. doi: 10.1007/BF02364990. Google Scholar [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. doi: 10.1093/imanum/dri016. Google Scholar [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. doi: 10.1137/S0036142903428967. Google Scholar [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. Google Scholar [29] J. Shen, Long time stability and convergence for fully discrete nonlinear Galerkin methods,, Appl. Anal., 38 (1990), 201. doi: 10.1080/00036819008839963. Google Scholar [30] P. Sobolevskii, Stabilization of viscoelastic fluid motion (Oldroyd's mathematical model),, Differential Integral Equations, 7 (1994), 1597. Google Scholar [31] P. Sobolevskii, Asymptotic of stable viscoelastic fluid motion (Oldroyd's mathematical model),, Math. Nachr., 177 (1996), 281. doi: 10.1002/mana.19961770116. Google Scholar [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. Google Scholar [33] R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", North-Holland, (1984). Google Scholar [34] R. Temam and X. Wang, Boundary layers asscociated with incompressible Navier-Stokes equations: the noncharacteristic boundary case,, J. Differential Equations, 179 (2002), 647. doi: 10.1006/jdeq.2001.4038. Google Scholar [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. doi: 10.1137/040618527. Google Scholar [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. doi: 10.1016/j.apm.2010.04.008. Google Scholar [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. doi: 10.1080/00207160.2010.534781. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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