# American Institute of Mathematical Sciences

November  2017, 22(9): 3421-3438. doi: 10.3934/dcdsb.2017173

## The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations

 School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

* Corresponding author: heyn@mail.xjtu.edu.cn

Received  April 2016 Revised  May 2017 Published  July 2017

Fund Project: The first author is supported by the NSF of China under grant No. 91630206. The second author is supported by the NSF of China under grant No. 11362021 and the Major Research and Development Program of China under grant No. 2016YFB0200901

In this paper, we present the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. The Galerkin mixed finite element satisfying inf-sup condition is used for the spatial discretization and the temporal treatment is implicit/explict scheme, which is Euler implicit scheme for the linear terms and explicit scheme for the nonlinear term. We prove that this method is almost unconditionally convergent and obtain the optimal $H^1-L^2$ error estimate of the numerical velocity-pressure under the hypothesis of $H^2$-regularity of the solution for the three dimensional nonstationary Navier-Stokes equations. Finally some numerical experiments are carried out to demonstrate the effectiveness of the method.

Citation: Jian Su, Yinnian He. The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3421-3438. doi: 10.3934/dcdsb.2017173
##### References:
 [1] R. A. Adams, Sobolev Space, Academic press, New York, 1975. Google Scholar [2] A. O. 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. Google Scholar [3] G. A. Baker, Galerkin Approximations for the Navier-Stokes Equations, manuscript, Harvard University, Cambridge, MA, 1976.Google Scholar [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. Google Scholar [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. Google Scholar [6] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. Google Scholar [7] W. E and J. G. Liu, Projection methods Ⅰ: Convergence and numerical boundary layers, SIAM J. Numer. Anal., 32 (1995), 1017-1057. doi: 10.1137/0732047. Google Scholar [8] G. Fairweather, H. P. Ma and W. W. Sun, Orthogonal spline collocation methods for the stream function-vorticity formulation of the Navier-Stokes equations, Numer. Methods for PDEs, 24 (2008), 449-464. doi: 10.1002/num.20269. Google Scholar [9] J. F. Gerbeau, Mathematical Methods for the Magnetohydrodynamics of Liquid Metals, Oxford univerisity press, Oxford, 2006. doi: 10.1093/acprof:oso/9780198566656.001.0001. Google Scholar [10] 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. Google Scholar [11] Y. N. He, Stability and error analysis for a spectral Galerkin method for the Navier-Stokes equations with $H^2$ or $H^1$ initial data, Numer. Methods for PDEs, 21 (2005), 875-904. doi: 10.1002/num.20065. Google Scholar [12] Y. N. He, Stability and error analysis for a spectral Galerkin method for the Navier-Stokes equations with with $L^2$ initial data, Numer. Methods for PDEs(24), (2008), 79-103. doi: 10.1002/num.20234. Google Scholar [13] 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. Google Scholar [14] 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. Google Scholar [15] 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. Google Scholar [16] 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), 1052-1078. doi: 10.1002/num.20077. Google Scholar [17] Y. N. He, K. M. Liu and W. W. Sun, Multi-level spectral Galerkin method for the Navier-Stokes equations Ⅰ: spatial discretization, Numer. Math., 101 (2005), 501-522. doi: 10.1007/s00211-005-0632-3. Google Scholar [18] Y. N. He, Y. P. Lin and W. W. Sun, Stabilized finite element methods for the nonstationary Navier-Stokes problem, Discrete and Continuous DynamicalSystems-Series B, 6 (2006), 41-68. Google Scholar [19] 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. Google Scholar [20] Y. N. He, Optimal error estimate of the penalty finite element method for the time-dependent Navier-Stokes problem, Mathmatics of Computation, 74 (2005), 1201-1216. doi: 10.1090/S0025-5718-05-01751-5. Google Scholar [21] 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. Google Scholar [22] Y. N. He, The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data, Mathmatics of Computation, 77 (2008), 2097-2124. doi: 10.1090/S0025-5718-08-02127-3. Google Scholar [23] 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. Google Scholar [24] 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. Google Scholar [25] 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. Google Scholar [26] H. Johnston and J. G. Liu, Accurate, stable and efficient Navier-Stokes slovers based on explicit treatment of the pressure term, J. Computational Physics, 199 (2004), 221-259. doi: 10.1016/j.jcp.2004.02.009. Google Scholar [27] 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. Google Scholar [28] H. P. Ma and W. W. Sun, Optimal Error Estimates of the Legendre Petro-Galerkin and pseudospectral methods for the generalized Korteweg-de Vries Equation, SIAM J. Numer. Anal., 39 (2001), 1380-1394. doi: 10.1137/S0036142900378327. Google Scholar [29] M. Marion and R. Temam, Navier-Stokes equations: Theory and approximation, in: Handbook of Numerical Analysis, North-Holland, Amsterdam, 4 (1998), 503-688. Google Scholar [30] R. H. Nochetto and J. H. Pyo, A finite element Gauge-Uzawa method Part Ⅰ: Navier-Stokes equations, SIAM J. Numer. Anal., 43 (2005), 1043-1068. doi: 10.1137/040609756. Google Scholar [31] J. Shen, Long time stability and convergence for fully discrete nonlinear Galerkin methods, Appl. Anal., 38 (1990), 201-229. doi: 10.1080/00036819008839963. Google Scholar [32] 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. Google Scholar [33] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, Studies in Mathematics and its Applications, Vol. 2. North-Holland Publishing Co. , Amsterdam-New York-Oxford, 1977. Google Scholar [34] F. Tone, Error analysis for a second scheme for the Navier-Stokes equations, Applied Numerical Mathematics, 50 (2004), 93-119. doi: 10.1016/j.apnum.2003.12.003. Google Scholar

show all references

##### References:
 [1] R. A. Adams, Sobolev Space, Academic press, New York, 1975. Google Scholar [2] A. O. 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. Google Scholar [3] G. A. Baker, Galerkin Approximations for the Navier-Stokes Equations, manuscript, Harvard University, Cambridge, MA, 1976.Google Scholar [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. Google Scholar [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. Google Scholar [6] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. Google Scholar [7] W. E and J. G. Liu, Projection methods Ⅰ: Convergence and numerical boundary layers, SIAM J. Numer. Anal., 32 (1995), 1017-1057. doi: 10.1137/0732047. Google Scholar [8] G. Fairweather, H. P. Ma and W. W. Sun, Orthogonal spline collocation methods for the stream function-vorticity formulation of the Navier-Stokes equations, Numer. Methods for PDEs, 24 (2008), 449-464. doi: 10.1002/num.20269. Google Scholar [9] J. F. Gerbeau, Mathematical Methods for the Magnetohydrodynamics of Liquid Metals, Oxford univerisity press, Oxford, 2006. doi: 10.1093/acprof:oso/9780198566656.001.0001. Google Scholar [10] 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. Google Scholar [11] Y. N. He, Stability and error analysis for a spectral Galerkin method for the Navier-Stokes equations with $H^2$ or $H^1$ initial data, Numer. Methods for PDEs, 21 (2005), 875-904. doi: 10.1002/num.20065. Google Scholar [12] Y. N. He, Stability and error analysis for a spectral Galerkin method for the Navier-Stokes equations with with $L^2$ initial data, Numer. Methods for PDEs(24), (2008), 79-103. doi: 10.1002/num.20234. Google Scholar [13] 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. Google Scholar [14] 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. Google Scholar [15] 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. Google Scholar [16] 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), 1052-1078. doi: 10.1002/num.20077. Google Scholar [17] Y. N. He, K. M. Liu and W. W. Sun, Multi-level spectral Galerkin method for the Navier-Stokes equations Ⅰ: spatial discretization, Numer. Math., 101 (2005), 501-522. doi: 10.1007/s00211-005-0632-3. Google Scholar [18] Y. N. He, Y. P. Lin and W. W. Sun, Stabilized finite element methods for the nonstationary Navier-Stokes problem, Discrete and Continuous DynamicalSystems-Series B, 6 (2006), 41-68. Google Scholar [19] 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. Google Scholar [20] Y. N. He, Optimal error estimate of the penalty finite element method for the time-dependent Navier-Stokes problem, Mathmatics of Computation, 74 (2005), 1201-1216. doi: 10.1090/S0025-5718-05-01751-5. Google Scholar [21] 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. Google Scholar [22] Y. N. He, The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data, Mathmatics of Computation, 77 (2008), 2097-2124. doi: 10.1090/S0025-5718-08-02127-3. Google Scholar [23] 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. Google Scholar [24] 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. Google Scholar [25] 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. Google Scholar [26] H. Johnston and J. G. Liu, Accurate, stable and efficient Navier-Stokes slovers based on explicit treatment of the pressure term, J. Computational Physics, 199 (2004), 221-259. doi: 10.1016/j.jcp.2004.02.009. Google Scholar [27] 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. Google Scholar [28] H. P. Ma and W. W. Sun, Optimal Error Estimates of the Legendre Petro-Galerkin and pseudospectral methods for the generalized Korteweg-de Vries Equation, SIAM J. Numer. Anal., 39 (2001), 1380-1394. doi: 10.1137/S0036142900378327. Google Scholar [29] M. Marion and R. Temam, Navier-Stokes equations: Theory and approximation, in: Handbook of Numerical Analysis, North-Holland, Amsterdam, 4 (1998), 503-688. Google Scholar [30] R. H. Nochetto and J. H. Pyo, A finite element Gauge-Uzawa method Part Ⅰ: Navier-Stokes equations, SIAM J. Numer. Anal., 43 (2005), 1043-1068. doi: 10.1137/040609756. Google Scholar [31] J. Shen, Long time stability and convergence for fully discrete nonlinear Galerkin methods, Appl. Anal., 38 (1990), 201-229. doi: 10.1080/00036819008839963. Google Scholar [32] 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. Google Scholar [33] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, Studies in Mathematics and its Applications, Vol. 2. North-Holland Publishing Co. , Amsterdam-New York-Oxford, 1977. Google Scholar [34] F. Tone, Error analysis for a second scheme for the Navier-Stokes equations, Applied Numerical Mathematics, 50 (2004), 93-119. doi: 10.1016/j.apnum.2003.12.003. Google Scholar
Comparison of the velocity ${u}$ and pressure $p$ with different time steps $\tau$ ($\nu=1.0$ and $h=1/24$)
Comparison of the error for the velocity ${u}$ and pressure $p$ with different times steps $\tau$
Comparison of the pressure $p$ at $T=1.0$ with different times steps ($\tau=0.1,0.05,0.01$)
Comparison of the velocity ${u}$ at $T=1.0$ in different $y$-plane($y=0.75,0.5,0.25)$)with different times steps $\tau=0.1(\text{top}),0.05(\text{middle}),0.01(\text{bottom})$
The norm $\|u^m_h\|_0$ of the Euler Explicit/Implicit scheme(T=6.0)
 1/h $\tau$ 0.2 0.3 0.4 0.5 0.75 1.0 16 0.641684e-5 0.642277e-5 0.642915e-5 0.643601e-5 0.645542e-5 0.64787e-5 24 0.655394e-5 0.65601e-5 0.656672e-5 0.65738e-5 0.659394e-5 0.661811e-5 32 0.660154e-5 0.660778e-5 0.661448e-5 0.662165e-5 0.664204e-5 0.666651e-5 40 0.662344e-5 0.662971e-5 0.663644e-5 0.664368e-5 0.666417e-5 0.66888e-5 48 0.663529e-5 0.664159e-5 0.664832e-5 0.665557e-5 0.667614e-5 0.670082e-5
 1/h $\tau$ 0.2 0.3 0.4 0.5 0.75 1.0 16 0.641684e-5 0.642277e-5 0.642915e-5 0.643601e-5 0.645542e-5 0.64787e-5 24 0.655394e-5 0.65601e-5 0.656672e-5 0.65738e-5 0.659394e-5 0.661811e-5 32 0.660154e-5 0.660778e-5 0.661448e-5 0.662165e-5 0.664204e-5 0.666651e-5 40 0.662344e-5 0.662971e-5 0.663644e-5 0.664368e-5 0.666417e-5 0.66888e-5 48 0.663529e-5 0.664159e-5 0.664832e-5 0.665557e-5 0.667614e-5 0.670082e-5
The norm $\|\nabla u^m_h\|_0$ of the Euler Explicit/Implicit scheme(T=6.0)
 1/h $\tau$ 0.2 0.3 0.4 0.5 0.75 1.0 16 0.528852e-4 0.529345e-4 0.529924e-4 0.530582e-4 0.532305e-4 0.53425e-4 24 0.535418e-4 0.535932e-4 0.536536e-4 0.537106e-4 0.538837e-4 0.540859e-4 32 0.537892e-4 0.538435e-4 0.539056e-4 0.539624e-4 0.54138e-4 0.543468e-4 40 0.539084e-4 0.539595e-4 0.54019e-4 0.540862e-4 0.542633e-4 0.544733e-4 48 0.539725e-4 0.540276e-4 0.540809e-4 0.541456e-4 0.543187e-4 0.545238e-4
 1/h $\tau$ 0.2 0.3 0.4 0.5 0.75 1.0 16 0.528852e-4 0.529345e-4 0.529924e-4 0.530582e-4 0.532305e-4 0.53425e-4 24 0.535418e-4 0.535932e-4 0.536536e-4 0.537106e-4 0.538837e-4 0.540859e-4 32 0.537892e-4 0.538435e-4 0.539056e-4 0.539624e-4 0.54138e-4 0.543468e-4 40 0.539084e-4 0.539595e-4 0.54019e-4 0.540862e-4 0.542633e-4 0.544733e-4 48 0.539725e-4 0.540276e-4 0.540809e-4 0.541456e-4 0.543187e-4 0.545238e-4
The norm $\|p^m_h\|_0$ of the Euler Explicit/Implicit scheme(T=6.0)
 1/h $\tau$ 0.2 0.3 0.4 0.5 0.75 1.0 16 0.0128003 0.0128003 0.0128003 0.0128003 0.0128003 0.0128003 24 0.0128003 0.0128003 0.0128003 0.0128003 0.0128003 0.0128003 32 0.0128002 0.0128002 0.0128003 0.0128002 0.0128003 0.0128003 40 0.0128002 0.0128002 0.0128002 0.0128003 0.0128003 0.0128003 48 0.0128002 0.0128002 0.0128002 0.0128002 0.0128002 0.0128002
 1/h $\tau$ 0.2 0.3 0.4 0.5 0.75 1.0 16 0.0128003 0.0128003 0.0128003 0.0128003 0.0128003 0.0128003 24 0.0128003 0.0128003 0.0128003 0.0128003 0.0128003 0.0128003 32 0.0128002 0.0128002 0.0128003 0.0128002 0.0128003 0.0128003 40 0.0128002 0.0128002 0.0128002 0.0128003 0.0128003 0.0128003 48 0.0128002 0.0128002 0.0128002 0.0128002 0.0128002 0.0128002
The convergence of the Euler Explicit/Implicit scheme
 h $\frac{\|u-u_h\|_{L^2}}{\|u\|_{L^2}}$ rate $\frac{\|\nabla(u-u_h)\|_{L^2}}{\|\nabla u\|_{L^2}}$ rate $\frac{\|p-p_h\|_{L^2}}{\|p\|_{L^2}}$ rate 1/16 0.402465e-1 / 0.187043 / 0.563091e-3 / 1/24 0.176129e-1 2.0381 0.118023 1.1356 0.268825e-3 1.8235 1/32 0.980015e-2 2.0378 0.0865438 1.0784 0.160528e-3 1.7922 1/40 0.621538e-2 2.0407 0.0684396 1.0518 0.108419e-3 1.7588 1/48 0.42794e-2 2.0470 0.0566459 1.0374 0.791043e-4 1.7290
 h $\frac{\|u-u_h\|_{L^2}}{\|u\|_{L^2}}$ rate $\frac{\|\nabla(u-u_h)\|_{L^2}}{\|\nabla u\|_{L^2}}$ rate $\frac{\|p-p_h\|_{L^2}}{\|p\|_{L^2}}$ rate 1/16 0.402465e-1 / 0.187043 / 0.563091e-3 / 1/24 0.176129e-1 2.0381 0.118023 1.1356 0.268825e-3 1.8235 1/32 0.980015e-2 2.0378 0.0865438 1.0784 0.160528e-3 1.7922 1/40 0.621538e-2 2.0407 0.0684396 1.0518 0.108419e-3 1.7588 1/48 0.42794e-2 2.0470 0.0566459 1.0374 0.791043e-4 1.7290
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