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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 |
References:
[1] |
R. A. Adams, Sobolev Spaces, Academic Press, New York, 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), 3-6. |
[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), 189-213.
doi: 10.1007/s002110050056. |
[4] |
P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. |
[5] |
V. Girault and P. A. Raviart, Finite Element Method for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, Berlin, Heidelberg, 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), 324-352. |
[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), 77-106.
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), 1263-1285.
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), 1052-1078.
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), 201-222.
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), 717-742. |
[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), 837-869.
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), 275-311.
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), 353-384.
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), 633-667.
doi: 10.1093/imanum/20.4.633. |
[18] |
D. D. Joseph, Fluid Dynamics of Viscoelastic Liquids, Springer Verlag, New York, 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, 14 (Russian), Trudy Mat. Inst. Steklov., 188 (1990), 59-87; Translated in Proc. Steklov Inst. Math., (1991), 73-108. |
[20] |
R. B. Kellogg and J. E. Osborn, A regularity result for the Stokes problem in a convex polygon, J. Functional Anal., 21 (1976), 397-431.
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), 2777-2788.
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), 348-365.
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), 23-70.
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, 13 (Russian), Trudy Mat. Inst. Steklov., 179 (1988), 126-164, 243; Translated in Proc. Steklov Inst. Math., (1989), 137-182. |
[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), Voprosy Kvant. Teor. Polya i Statist. Fiz. 10, 101-121, 183-184; Translation in J. Soviet Math., 62 (1992), 3004-3016.
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), 750-782.
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), 804-825.
doi: 10.1137/S0036142903428967. |
[28] |
J. Shen, Long time stability and convergence for fully discrete nonlinear Galerkin methods, Appl. Anal., 38 (1990), 201-229.
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), 96-110.
doi: 10.1016/j.jmaa.2014.12.020. |
[30] |
R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, 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), 665-684.
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), 2199-2220.
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), 657-677. |
[34] |
K. Wang, Z. Si and Y. Yang, Stabilized finite element method for the viscoelastic Oldroyd fluid flows, Numer. Algorithms, 60 (2012), 75-100.
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), 1551-1573.
doi: 10.3934/dcdsb.2012.17.1551. |
[36] |
W. L. Wilkinson, Non-Newtonian Fluids: Fluid Mechanics, Mixing and Heat Transfer, Pergamon Press, Oxford, UK, 1960. |
show all references
References:
[1] |
R. A. Adams, Sobolev Spaces, Academic Press, New York, 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), 3-6. |
[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), 189-213.
doi: 10.1007/s002110050056. |
[4] |
P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. |
[5] |
V. Girault and P. A. Raviart, Finite Element Method for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, Berlin, Heidelberg, 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), 324-352. |
[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), 77-106.
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), 1263-1285.
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), 1052-1078.
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), 201-222.
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), 717-742. |
[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), 837-869.
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), 275-311.
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), 353-384.
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), 633-667.
doi: 10.1093/imanum/20.4.633. |
[18] |
D. D. Joseph, Fluid Dynamics of Viscoelastic Liquids, Springer Verlag, New York, 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, 14 (Russian), Trudy Mat. Inst. Steklov., 188 (1990), 59-87; Translated in Proc. Steklov Inst. Math., (1991), 73-108. |
[20] |
R. B. Kellogg and J. E. Osborn, A regularity result for the Stokes problem in a convex polygon, J. Functional Anal., 21 (1976), 397-431.
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), 2777-2788.
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), 348-365.
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), 23-70.
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, 13 (Russian), Trudy Mat. Inst. Steklov., 179 (1988), 126-164, 243; Translated in Proc. Steklov Inst. Math., (1989), 137-182. |
[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), Voprosy Kvant. Teor. Polya i Statist. Fiz. 10, 101-121, 183-184; Translation in J. Soviet Math., 62 (1992), 3004-3016.
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), 750-782.
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), 804-825.
doi: 10.1137/S0036142903428967. |
[28] |
J. Shen, Long time stability and convergence for fully discrete nonlinear Galerkin methods, Appl. Anal., 38 (1990), 201-229.
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), 96-110.
doi: 10.1016/j.jmaa.2014.12.020. |
[30] |
R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, 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), 665-684.
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), 2199-2220.
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), 657-677. |
[34] |
K. Wang, Z. Si and Y. Yang, Stabilized finite element method for the viscoelastic Oldroyd fluid flows, Numer. Algorithms, 60 (2012), 75-100.
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), 1551-1573.
doi: 10.3934/dcdsb.2012.17.1551. |
[36] |
W. L. Wilkinson, Non-Newtonian Fluids: Fluid Mechanics, Mixing and Heat Transfer, Pergamon Press, Oxford, UK, 1960. |
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