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Long time behavior of fractional impulsive stochastic differential equations with infinite delay
H2-stability of some second order fully discrete schemes for the Navier-Stokes equations
1. | School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China |
2. | College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China |
3. | Department of Mathematics, School of Arts and Sciences, Shaanxi University of Science and Technology, Xi'an 710021, China |
4. | Department of Mathematics, Baoji University of Arts and Sciences, Baoji 721013, China |
This paper considers the $H^2$-stability results for the second order fully discrete schemes based on the mixed finite element method for the 2D time-dependent Navier-Stokes equations with the initial data $u_0∈ H^α, $ where $α = 0, ~1$ and 2. A mixed finite element method is used to the spatial discretization of the Navier-Stokes equations, and the temporal treatments of the spatial discrete Navier-Stokes equations are the second order semi-implicit, implicit/explict and explicit schemes. The $H^2$-stability results of the schemes are provided, where the second order semi-implicit and implicit/explicit schemes are almost unconditionally $H^2$-stable, the second order explicit scheme is conditionally $H^2$-stable in the case of $\alpha = 2$, and the semi-implicit, implicit/explicit and explicit schemes are conditionally $H^2$-stable in the case of $\alpha = 1, ~0$. Finally, some numerical tests are made to verify the above theoretical results.
References:
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R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. |
[2] |
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. |
[3] |
D. N. Arnold, F. Brezzi and M. Fortin,
A stable finite element for the Stokes equations, Calcolo, 21 (1984), 337-344.
doi: 10.1007/BF02576171. |
[4] |
G. A. Baker, V. A. Dougalis and O. A. Karakashian,
On a high order accurate fully discrete Galerkin approximation to the Navier-Stokes equations, Math. Comp., 39 (1982), 339-375.
doi: 10.1090/S0025-5718-1982-0669634-0. |
[5] |
J. Bercovier and O. Pironneau,
Error estimates for finite element solution of the Stokes problem in the primitive variables, Numer. Math., 33 (1979), 211-224.
doi: 10.1007/BF01399555. |
[6] |
P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. |
[7] |
J. F. Gerbeau, C. Le Bris and T. Lelièvre,
Mathematical Method for the Magnetohydrodynamics of Liquid Metals, Oxford University Press, Oxford, 2006.
doi: 10.1093/acprof:oso/9780198566656.001.0001. |
[8] |
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. |
[9] |
Y. N. He and K. T. Li,
Nonlinear Galerkin method and two-step method for the Navier-Stokes equations, Numer. Methods for PDEs, 12 (1996), 283-305.
doi: 10.1002/(SICI)1098-2426(199605)12:3<283::AID-NUM1>3.0.CO;2-K. |
[10] |
Y. N. He and K. T. 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. |
[11] |
Y. N. 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. |
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Y. N. He, H. L. Miao, R. M. M. Mattheij and Z. X. Chen,
Numerical analysis of a modified finite element nonlinear Galerkin method, Numer. Math., 97 (2004), 725-756.
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Y. N. He,
Stability and error analysis for a spectral Galerkin method for the Navier-Stokes equations with H2 or H1 initial data, Numer. Methods for PDEs, 21 (2005), 875-904.
doi: 10.1002/num.20065. |
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Y. N. He,
Optimal error estimate of the penalty finite element method for the time-dependent Navier-Stokes problem, Math. Comp., 74 (2005), 1201-1216.
doi: 10.1090/S0025-5718-05-01751-5. |
[15] |
Y. N. He and K. M. Liu,
A multi-level finite element method for the time-dependent Navier-Stokes equations, Numer. Methods for PDEs, 21 (2005), 1052-1068.
|
[16] |
Y. N. He and W. W. Sun,
Stability and convegence 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] |
Y. N. He,
Stability and error analysis for a spectral Galerkin method for the Navier-Stokes equations with with L2 initial data, Numer. Methods for PDEs, 24 (2008), 79-103.
doi: 10.1002/num.20234. |
[18] |
Y. N. He,
Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data, Math. Comp., 77 (2008), 2097-2124.
doi: 10.1090/S0025-5718-08-02127-3. |
[19] |
Y. N. He,
Stability and convegence of the Crank-Nicolson/Adams-Bashforth scheme for the time-dependent Navier-Stokes equations with non-smooth initial data, Numer. Methods for PDEs, 28 (2012), 155-187.
doi: 10.1002/num.20613. |
[20] |
Y. N. He, P. Z. Huang and X. L. Feng,
H2-stability of the first order finite element fully discrete schemes for the 2D time-dependent Navier-Stokes equations with smooth and non-smooth initial data, J. Sci. Comput., 62 (2015), 230-264.
doi: 10.1007/s10915-014-9854-9. |
[21] |
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. |
[22] |
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. |
[23] |
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. |
[24] |
H. Johnston and J. G. Liu,
Accurate, stable and efficient Navier-Stokes slovers based on explicit treatment of the pressure term, J. Comput. Phys., 199 (2004), 221-259.
doi: 10.1016/j.jcp.2004.02.009. |
[25] |
J. Kim and P. Moin,
Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59 (1985), 308-323.
doi: 10.1016/0021-9991(85)90148-2. |
[26] |
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. |
[27] |
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. |
[28] |
M. Marion and R. Temam, Navier-Stokes equations: Theory and approximation, in: Handbook of Numerical Analysis, Vol. Ⅵ, pp. 503–688, North-Holland, Amsterdam, 1998. |
[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] |
J. C. Simo and F. Armero,
Unconditional stability and long-term behavior of transient algorithms for the incompressible Navier-Stokes and Euler equations, Comput. Methods Appl. Mech. Engrg., 111 (1994), 111-154.
doi: 10.1016/0045-7825(94)90042-6. |
[31] |
R. Temam,
Navier-Stokes Equations, Theory and Numerical Analysis, 3rd ed., North-Holland, Amsterdam, 1984. |
[32] |
F. Tone,
Error analysis for a second scheme for the Navier-Stokes equations, Appl. Numer. Math., 50 (2004), 93-119.
doi: 10.1016/j.apnum.2003.12.003. |
show all references
References:
[1] |
R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. |
[2] |
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. |
[3] |
D. N. Arnold, F. Brezzi and M. Fortin,
A stable finite element for the Stokes equations, Calcolo, 21 (1984), 337-344.
doi: 10.1007/BF02576171. |
[4] |
G. A. Baker, V. A. Dougalis and O. A. Karakashian,
On a high order accurate fully discrete Galerkin approximation to the Navier-Stokes equations, Math. Comp., 39 (1982), 339-375.
doi: 10.1090/S0025-5718-1982-0669634-0. |
[5] |
J. Bercovier and O. Pironneau,
Error estimates for finite element solution of the Stokes problem in the primitive variables, Numer. Math., 33 (1979), 211-224.
doi: 10.1007/BF01399555. |
[6] |
P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. |
[7] |
J. F. Gerbeau, C. Le Bris and T. Lelièvre,
Mathematical Method for the Magnetohydrodynamics of Liquid Metals, Oxford University Press, Oxford, 2006.
doi: 10.1093/acprof:oso/9780198566656.001.0001. |
[8] |
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. |
[9] |
Y. N. He and K. T. Li,
Nonlinear Galerkin method and two-step method for the Navier-Stokes equations, Numer. Methods for PDEs, 12 (1996), 283-305.
doi: 10.1002/(SICI)1098-2426(199605)12:3<283::AID-NUM1>3.0.CO;2-K. |
[10] |
Y. N. He and K. T. 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. |
[11] |
Y. N. 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. |
[12] |
Y. N. He, H. L. Miao, R. M. M. Mattheij and Z. X. Chen,
Numerical analysis of a modified finite element nonlinear Galerkin method, Numer. Math., 97 (2004), 725-756.
doi: 10.1007/s00211-003-0516-3. |
[13] |
Y. N. He,
Stability and error analysis for a spectral Galerkin method for the Navier-Stokes equations with H2 or H1 initial data, Numer. Methods for PDEs, 21 (2005), 875-904.
doi: 10.1002/num.20065. |
[14] |
Y. N. He,
Optimal error estimate of the penalty finite element method for the time-dependent Navier-Stokes problem, Math. Comp., 74 (2005), 1201-1216.
doi: 10.1090/S0025-5718-05-01751-5. |
[15] |
Y. N. He and K. M. Liu,
A multi-level finite element method for the time-dependent Navier-Stokes equations, Numer. Methods for PDEs, 21 (2005), 1052-1068.
|
[16] |
Y. N. He and W. W. Sun,
Stability and convegence 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] |
Y. N. He,
Stability and error analysis for a spectral Galerkin method for the Navier-Stokes equations with with L2 initial data, Numer. Methods for PDEs, 24 (2008), 79-103.
doi: 10.1002/num.20234. |
[18] |
Y. N. He,
Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data, Math. Comp., 77 (2008), 2097-2124.
doi: 10.1090/S0025-5718-08-02127-3. |
[19] |
Y. N. He,
Stability and convegence of the Crank-Nicolson/Adams-Bashforth scheme for the time-dependent Navier-Stokes equations with non-smooth initial data, Numer. Methods for PDEs, 28 (2012), 155-187.
doi: 10.1002/num.20613. |
[20] |
Y. N. He, P. Z. Huang and X. L. Feng,
H2-stability of the first order finite element fully discrete schemes for the 2D time-dependent Navier-Stokes equations with smooth and non-smooth initial data, J. Sci. Comput., 62 (2015), 230-264.
doi: 10.1007/s10915-014-9854-9. |
[21] |
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. |
[22] |
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. |
[23] |
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. |
[24] |
H. Johnston and J. G. Liu,
Accurate, stable and efficient Navier-Stokes slovers based on explicit treatment of the pressure term, J. Comput. Phys., 199 (2004), 221-259.
doi: 10.1016/j.jcp.2004.02.009. |
[25] |
J. Kim and P. Moin,
Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59 (1985), 308-323.
doi: 10.1016/0021-9991(85)90148-2. |
[26] |
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. |
[27] |
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. |
[28] |
M. Marion and R. Temam, Navier-Stokes equations: Theory and approximation, in: Handbook of Numerical Analysis, Vol. Ⅵ, pp. 503–688, North-Holland, Amsterdam, 1998. |
[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] |
J. C. Simo and F. Armero,
Unconditional stability and long-term behavior of transient algorithms for the incompressible Navier-Stokes and Euler equations, Comput. Methods Appl. Mech. Engrg., 111 (1994), 111-154.
doi: 10.1016/0045-7825(94)90042-6. |
[31] |
R. Temam,
Navier-Stokes Equations, Theory and Numerical Analysis, 3rd ed., North-Holland, Amsterdam, 1984. |
[32] |
F. Tone,
Error analysis for a second scheme for the Navier-Stokes equations, Appl. Numer. Math., 50 (2004), 93-119.
doi: 10.1016/j.apnum.2003.12.003. |



0.005 | 0.01 | 0.02 | 0.05 | 0.5 | |
1.753E |
1.752E |
1.752E |
1.842E |
5.788E |
|
7.330E |
7.330E |
7.331E |
8.480E |
8.917E |
|
1.166E |
1.839E |
3.357E |
8.126E |
8.204E |
0.005 | 0.01 | 0.02 | 0.05 | 0.5 | |
1.753E |
1.752E |
1.752E |
1.842E |
5.788E |
|
7.330E |
7.330E |
7.331E |
8.480E |
8.917E |
|
1.166E |
1.839E |
3.357E |
8.126E |
8.204E |
0.005 | 0.01 | 0.02 | 0.05 | 0.5 | |
1.753E |
1.752E |
1.752E |
1.842E |
5.925E |
|
7.330E |
7.330E |
7.331E |
8.480E |
9.291E |
|
1.166E |
1.839E |
3.357E |
8.126E |
8.204E |
0.005 | 0.01 | 0.02 | 0.05 | 0.5 | |
1.753E |
1.752E |
1.752E |
1.842E |
5.925E |
|
7.330E |
7.330E |
7.331E |
8.480E |
9.291E |
|
1.166E |
1.839E |
3.357E |
8.126E |
8.204E |
0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 | |
2.935 | 0.579 | 0.1638 | 0.873 | 1.423 | 3.564 | 2.019 | 1.740 | 0.082 | 0.140 | |
3.512 | 1.641 | 0.379 | 0.475 | 1.537 | 2.391 | 2.158 | 1.048 | 0.133 | 0.332 | |
1.886 | 1.035 | 0.038 | 0.467 | 1.754 | 2.654 | 2.378 | 1.158 | 0.121 | 0.218 | |
2.152 | 0.792 | 0.014 | 0.469 | 1.739 | 2.655 | 2.376 | 1.157 | 0.120 | 0.218 | |
2.086 | 0.791 | 0.014 | 0.469 | 1.739 | 2.655 | 2.376 | 1.157 | 0.120 | 0.218 |
0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 | |
2.935 | 0.579 | 0.1638 | 0.873 | 1.423 | 3.564 | 2.019 | 1.740 | 0.082 | 0.140 | |
3.512 | 1.641 | 0.379 | 0.475 | 1.537 | 2.391 | 2.158 | 1.048 | 0.133 | 0.332 | |
1.886 | 1.035 | 0.038 | 0.467 | 1.754 | 2.654 | 2.378 | 1.158 | 0.121 | 0.218 | |
2.152 | 0.792 | 0.014 | 0.469 | 1.739 | 2.655 | 2.376 | 1.157 | 0.120 | 0.218 | |
2.086 | 0.791 | 0.014 | 0.469 | 1.739 | 2.655 | 2.376 | 1.157 | 0.120 | 0.218 |
0.0005 | 0.005 | 0.05 | 0.5 | |
1.894E |
1.894E |
1.980E |
4.376E |
|
6.442E |
6.442E |
7.719E |
8.131E |
|
8.554E |
1.355E |
8.366E |
8.208E |
0.0005 | 0.005 | 0.05 | 0.5 | |
1.894E |
1.894E |
1.980E |
4.376E |
|
6.442E |
6.442E |
7.719E |
8.131E |
|
8.554E |
1.355E |
8.366E |
8.208E |
0.0005 | 0.005 | 0.05 | 0.5 | |
1.894E |
1.894E |
1.980E |
4.564E |
|
6.442E |
6.442E |
7.718E |
8.541E |
|
8.554E |
1.355E |
8.366E |
8.208E |
0.0005 | 0.005 | 0.05 | 0.5 | |
1.894E |
1.894E |
1.980E |
4.564E |
|
6.442E |
6.442E |
7.718E |
8.541E |
|
8.554E |
1.355E |
8.366E |
8.208E |
0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 | |
2.627 | 0.516 | 0.156 | 0.798 | 1.248 | 3.217 | 1.768 | 1.587 | 0.073 | 0.120 | |
3.219 | 1.523 | 0.373 | 0.437 | 1.370 | 2.125 | 1.919 | 0.934 | 0.124 | 0.303 | |
1.856 | 0.704 | 0.012 | 0.417 | 1.546 | 2.362 | 2.114 | 1.030 | 0.112 | 0.194 | |
1.857 | 0.704 | 0.012 | 0.417 | 1.546 | 2.362 | 2.114 | 1.030 | 0.107 | 0.194 |
0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 | |
2.627 | 0.516 | 0.156 | 0.798 | 1.248 | 3.217 | 1.768 | 1.587 | 0.073 | 0.120 | |
3.219 | 1.523 | 0.373 | 0.437 | 1.370 | 2.125 | 1.919 | 0.934 | 0.124 | 0.303 | |
1.856 | 0.704 | 0.012 | 0.417 | 1.546 | 2.362 | 2.114 | 1.030 | 0.112 | 0.194 | |
1.857 | 0.704 | 0.012 | 0.417 | 1.546 | 2.362 | 2.114 | 1.030 | 0.107 | 0.194 |
0.0005 | 0.005 | 0.05 | 0.5 | |
2.979E |
2.979E |
2.977E |
3.768E |
|
3.474E |
3.474E |
3.492E |
3.589E |
|
2.847E |
2.204E |
5.737E |
8.168E |
0.0005 | 0.005 | 0.05 | 0.5 | |
2.979E |
2.979E |
2.977E |
3.768E |
|
3.474E |
3.474E |
3.492E |
3.589E |
|
2.847E |
2.204E |
5.737E |
8.168E |
0.0005 | 0.005 | 0.05 | 0.5 | |
2.979E |
2.979E |
2.977E |
3.768E |
|
3.474E |
3.474E |
3.492E |
3.599E |
|
2.847E |
2.204E |
5.737E |
8.168E |
0.0005 | 0.005 | 0.05 | 0.5 | |
2.979E |
2.979E |
2.977E |
3.768E |
|
3.474E |
3.474E |
3.492E |
3.599E |
|
2.847E |
2.204E |
5.737E |
8.168E |
0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 | |
30.267 | 4.347 | 1.423 | 9.108 | 12.374 | 36.539 | 18.046 | 17.898 | 0.223 | 0.714 | |
20.143 | 7.701 | 0.409 | 5.202 | 17.862 | 26.862 | 23.947 | 11.731 | 1.319 | 2.133 | |
20.510 | 7.750 | 0.128 | 4.653 | 17.166 | 26.162 | 23.370 | 11.358 | 1.168 | 2.170 | |
20.512 | 7.750 | 0.128 | 4.653 | 17.166 | 26.162 | 23.370 | 11.357 | 1.168 | 2.170 |
0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 | |
30.267 | 4.347 | 1.423 | 9.108 | 12.374 | 36.539 | 18.046 | 17.898 | 0.223 | 0.714 | |
20.143 | 7.701 | 0.409 | 5.202 | 17.862 | 26.862 | 23.947 | 11.731 | 1.319 | 2.133 | |
20.510 | 7.750 | 0.128 | 4.653 | 17.166 | 26.162 | 23.370 | 11.358 | 1.168 | 2.170 | |
20.512 | 7.750 | 0.128 | 4.653 | 17.166 | 26.162 | 23.370 | 11.357 | 1.168 | 2.170 |
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