• Previous Article
    Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$
  • DCDS-B Home
  • This Issue
  • Next Article
    Local well-posedness and small Deborah limit of a molecule-based $Q$-tensor system
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).

[1]

Sondre Tesdal Galtung. A convergent Crank-Nicolson Galerkin scheme for the Benjamin-Ono equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1243-1268. doi: 10.3934/dcds.2018051

[2]

Dongho Kim, Eun-Jae Park. Adaptive Crank-Nicolson methods with dynamic finite-element spaces for parabolic problems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 873-886. doi: 10.3934/dcdsb.2008.10.873

[3]

Alexander Zlotnik. The Numerov-Crank-Nicolson scheme on a non-uniform mesh for the time-dependent Schrödinger equation on the half-axis. Kinetic & Related Models, 2015, 8 (3) : 587-613. doi: 10.3934/krm.2015.8.587

[4]

Nicolas Crouseilles, Mohammed Lemou, SV Raghurama Rao, Ankit Ruhi, Muddu Sekhar. Asymptotic preserving scheme for a kinetic model describing incompressible fluids. Kinetic & Related Models, 2016, 9 (1) : 51-74. doi: 10.3934/krm.2016.9.51

[5]

Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie. A combined finite volume - finite element scheme for a dispersive shallow water system. Networks & Heterogeneous Media, 2016, 11 (1) : 1-27. doi: 10.3934/nhm.2016.11.1

[6]

François Alouges. A new finite element scheme for Landau-Lifchitz equations. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 187-196. doi: 10.3934/dcdss.2008.1.187

[7]

Daoyuan Fang, Ting Zhang, Ruizhao Zi. Dispersive effects of the incompressible viscoelastic fluids. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5261-5295. doi: 10.3934/dcds.2018233

[8]

Matthias Hieber. Remarks on the theory of Oldroyd-B fluids in exterior domains. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1307-1313. doi: 10.3934/dcdss.2013.6.1307

[9]

R. Ryham, Chun Liu, Zhi-Qiang Wang. On electro-kinetic fluids: One dimensional configurations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 357-371. doi: 10.3934/dcdsb.2006.6.357

[10]

Wen Li, Song Wang, Volker Rehbock. A 2nd-order one-point numerical integration scheme for fractional ordinary differential equations. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 273-287. doi: 10.3934/naco.2017018

[11]

Giovambattista Amendola, Sandra Carillo, John Murrough Golden, Adele Manes. Viscoelastic fluids: Free energies, differential problems and asymptotic behaviour. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1815-1835. doi: 10.3934/dcdsb.2014.19.1815

[12]

Colette Guillopé, Zaynab Salloum, Raafat Talhouk. Regular flows of weakly compressible viscoelastic fluids and the incompressible limit. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1001-1028. doi: 10.3934/dcdsb.2010.14.1001

[13]

Wei Qu, Siu-Long Lei, Seak-Weng Vong. A note on the stability of a second order finite difference scheme for space fractional diffusion equations. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 317-325. doi: 10.3934/naco.2014.4.317

[14]

Francis Filbet, Roland Duclous, Bruno Dubroca. Analysis of a high order finite volume scheme for the 1D Vlasov-Poisson system. Discrete & Continuous Dynamical Systems - S, 2012, 5 (2) : 283-305. doi: 10.3934/dcdss.2012.5.283

[15]

Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59

[16]

Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133

[17]

François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221

[18]

Bertram Düring, Daniel Matthes, Josipa Pina Milišić. A gradient flow scheme for nonlinear fourth order equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 935-959. doi: 10.3934/dcdsb.2010.14.935

[19]

Giulio G. Giusteri, Alfredo Marzocchi, Alessandro Musesti. Nonlinear free fall of one-dimensional rigid bodies in hyperviscous fluids. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2145-2157. doi: 10.3934/dcdsb.2014.19.2145

[20]

Kun Wang, Yinnian He, Yueqiang Shang. Fully discrete finite element method for the viscoelastic fluid motion equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 665-684. doi: 10.3934/dcdsb.2010.13.665

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (3)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]