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June  2021, 26(6): 3119-3142. doi: 10.3934/dcdsb.2020222

A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: Yueqiang Shang

Received  January 2020 Revised  May 2020 Published  June 2021 Early access  July 2020

Fund Project: The first author is supported by the Natural Science Foundation of China (No. 11361016), the Basic and Frontier Explore Program of Chongqing Municipality, China (No. cstc2018jcyjAX0305), and Fundamental Research Funds for the Central Universities (No. XDJK2018B032)

A postprocessed mixed finite element method based on a subgrid model is presented for the simulation of time-dependent incompressible Navier-Stokes equations. This method consists of two steps: the first step is to solve a subgrid stabilized nonlinear Navier-Stokes system on a coarse grid to obtain an approximate solution $ u_{H}(x,T) $ at the final time $ T $, and the second step is to postprocess $ u_{H}(x,T) $ by solving a stabilized Stokes problem on a finer grid or by higher-order finite element elements defined on the same coarse grid. Stability of the method and error estimates of the processing solution are analyzed. Numerical results on an example with known analytic solution and the flow around a circular cylinder are given to verify the theoretical predictions and demonstrate the effectiveness of the proposed method.

Citation: Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222
References:
[1]

H. AbboudV. Girault and T. Sayah, A second order accuracy for a full discretized time-dependent Navier-Stokes equations by a two-grid scheme, Numer. Math., 114 (2009), 189-231.  doi: 10.1007/s00211-009-0251-5.

[2]

R. A. Adams, Sobolev Spaces., Academic Press Inc., New York, 1975.

[3]

B. AyusoB. García-Archilla and J. Novo, The postprocessed mixed finite-element method for the Navier-Stokes equations, SIAM J. Numer. Anal., 43 (2005), 1091-1111.  doi: 10.1137/040602821.

[4]

B. AyusoJ. de Frutos and J. Novo, Improving the accuracy of the mini-element approximation to Navier-Stokes equations, IMA J. Numer. Anal., 27 (2007), 198-218.  doi: 10.1093/imanum/drl010.

[5]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978.

[6]

J. de FrutosB. García-ArchillaV. John and J. Novo, Analysis of the grad-div stabilization for the time-dependent Navier-Stokes equations with inf-sup stable finite elements, Adv. Comput. Math., 44 (2018), 195-225.  doi: 10.1007/s10444-017-9540-1.

[7]

J. de FrutosB. García-Archilla and J. Novo, Static two-grid mixed finite-element approximations to the Navier-Stokes equations, J. Sci. Comput., 52 (2012), 619-637.  doi: 10.1007/s10915-011-9562-7.

[8]

F. Durango and J. Novo, Two-grid mixed finite-element approximations to the Navier-Stokes equations based on a Newton-type step, J. Sci. Comput., 74 (2018), 456-473.  doi: 10.1007/s10915-017-0447-2.

[9] H. C. ElmanD. J. Silvester and A. J. Wathen, Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, Oxford University Press, New York, 2005. 
[10]

B. García-ArchillaJ. Novo and E. S. Titi, Postprocessing the Galerkin method: A novel approach to approximate iunertial manifolds, SIAM J. Numer. Anal., 35 (1998), 941-972.  doi: 10.1137/S0036142995296096.

[11]

B. García-ArchillaJ. Novo and E. S. Titi, An approximate inertial manifold approach to postprocessing Galerkin methods for the Navier-Stokes equations, Math. Comp., 68 (1999), 893-911.  doi: 10.1090/S0025-5718-99-01057-1.

[12]

B. García-Archilla and E. S. Titi, Postprocessing the Galerkin method: the finite-element case, SIAM J. Numer. Anal., 37 (2000), 470-499.  doi: 10.1137/S0036142998335893.

[13]

V. Girault and J.-L. Lions, Two-grid finite element scheme for the transient Navier-Stokes problem, M2AN Math. Model. Numer. Anal., 35 (2001), 945-980.  doi: 10.1051/m2an:2001145.

[14]

V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, Berlin Heidelberg, 1986. doi: 10.1007/978-3-642-61623-5.

[15]

R. Glowinski, Finite Element Methods for Incompressible Viscous Flow, Handbook of numerical analysis, Vol. IX, 3–1176, Handb. Numer. Anal., IX, North-Holland, Amsterdam, 2003.

[16]

J.-L. Guermond, Stabilization of Galerkin approximations of transport equations by subgrid modeling, M2AN Math. Model. Numer. Anal., 33 (1999), 1293-1316.  doi: 10.1051/m2an:1999145.

[17]

J.-L. Guermond, Subgrid stabilization of Galerkin approximations of linear contraction semi-groups of class $C^0$ in Hilbert spaces, Numer. Meth. PDEs., 17 (2001), 1-25. 

[18]

J.-L. Guermond, A. Marra and L. Quartapelle, Subgrid stabilized projection method for 2D unsteady flows at high Reynolds numbers, Comput. Meth. Appl. Mech. Engrg., 195)(2006), 5857–5876. doi: 10.1016/j.cma.2005.08.016.

[19]

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.

[20]

Y. N. He, A two-level finite element Galerkin method for the nonstationary Navier-Stokes equations, I: Spatial discretization, J. Comput. Math., 22 (2004), 21-32. 

[21]

Y. N. He and K. M. Liu, A multi-level finite element method in space-time for the Navier-Stokes equations, Numer. Meth. PDEs., 21 (2005), 1052-1078. 

[22]

Y. N. HeH. L. Miao and C. F. Ren, A two-level finite element Galerkin method for the nonstationary Navier-Stokes equations, Ⅱ: Time discretization, J. Comput. Math., 22 (2004), 33-54. 

[23]

Y. N. He and W. 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.

[24]

F. Hecht, New development in Freefem++, J. Numer. Math., 20 (2012), 251-266.  doi: 10.1515/jnum-2012-0013.

[25]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981.

[26]

T. J. R. Hughes, Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid-scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Engrg., 127 (1995), 387-401.  doi: 10.1016/0045-7825(95)00844-9.

[27]

T. J. R. HughesL. Mazzei and K. E. Jansen, Large eddy simulation and the variational multiscale method, Comput. Vis. Sci., 3 (2000), 47-59.  doi: 10.1007/s007910050051.

[28]

V. John, Finite Element Methods for Incompressible Flow Problems, Springer Series in Computational Mathematics, 51. Springer, Cham, 2016. doi: 10.1007/978-3-319-45750-5.

[29]

A. Labovschii, A defect correction method for the time-dependent Navier-Stokes equations, Numer. Meth. PDEs., 25 (2009), 1-25.  doi: 10.1002/num.20329.

[30]

W. Layton, A connection between subgrid scale eddy viscosity and mixed methods, Appl. Math. Comput., 133 (2002), 147-157.  doi: 10.1016/S0096-3003(01)00228-4.

[31]

W. LaytonH. K. Lee and J. Peterson, A defect-correction method for the incompressible Navier-Stokes equations, Appl. Math. Comput., 129 (2002), 1-19.  doi: 10.1016/S0096-3003(01)00026-1.

[32]

M. A. Olshanskii, Two-level method and some a priori estimates in unsteady Navier-Stokes calculations, J. Comput. Appl. Math., 104 (1999), 173-191.  doi: 10.1016/S0377-0427(99)00056-4.

[33]

Y. Q. Shang, A two-level subgrid stabilized Oseen iterative method for the steady Navier-Stokes equations, J. Comput. Phys., 233 (2013), 210-226.  doi: 10.1016/j.jcp.2012.08.024.

[34]

J. Smagorinsky, General circulation experiments with the primitive equations, I: The basic experiments, Mon. Wea. Rev., 91 (1963), 99-164.  doi: 10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2.

[35]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam 1984.

[36]

K. Wang, A new defect correction method for the Navier-Stokes equations at high Reynolds numbers, Appl. Math. Comput., 216 (2010), 3252-3264.  doi: 10.1016/j.amc.2010.04.050.

[37]

Y. Zhang and Y. N. He, Assessment of subgrid-scale models for the incompressible Navier-Stokes equations, J. Comput. Appl. Math., 234 (2010), 593-604.  doi: 10.1016/j.cam.2009.12.051.

show all references

References:
[1]

H. AbboudV. Girault and T. Sayah, A second order accuracy for a full discretized time-dependent Navier-Stokes equations by a two-grid scheme, Numer. Math., 114 (2009), 189-231.  doi: 10.1007/s00211-009-0251-5.

[2]

R. A. Adams, Sobolev Spaces., Academic Press Inc., New York, 1975.

[3]

B. AyusoB. García-Archilla and J. Novo, The postprocessed mixed finite-element method for the Navier-Stokes equations, SIAM J. Numer. Anal., 43 (2005), 1091-1111.  doi: 10.1137/040602821.

[4]

B. AyusoJ. de Frutos and J. Novo, Improving the accuracy of the mini-element approximation to Navier-Stokes equations, IMA J. Numer. Anal., 27 (2007), 198-218.  doi: 10.1093/imanum/drl010.

[5]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978.

[6]

J. de FrutosB. García-ArchillaV. John and J. Novo, Analysis of the grad-div stabilization for the time-dependent Navier-Stokes equations with inf-sup stable finite elements, Adv. Comput. Math., 44 (2018), 195-225.  doi: 10.1007/s10444-017-9540-1.

[7]

J. de FrutosB. García-Archilla and J. Novo, Static two-grid mixed finite-element approximations to the Navier-Stokes equations, J. Sci. Comput., 52 (2012), 619-637.  doi: 10.1007/s10915-011-9562-7.

[8]

F. Durango and J. Novo, Two-grid mixed finite-element approximations to the Navier-Stokes equations based on a Newton-type step, J. Sci. Comput., 74 (2018), 456-473.  doi: 10.1007/s10915-017-0447-2.

[9] H. C. ElmanD. J. Silvester and A. J. Wathen, Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, Oxford University Press, New York, 2005. 
[10]

B. García-ArchillaJ. Novo and E. S. Titi, Postprocessing the Galerkin method: A novel approach to approximate iunertial manifolds, SIAM J. Numer. Anal., 35 (1998), 941-972.  doi: 10.1137/S0036142995296096.

[11]

B. García-ArchillaJ. Novo and E. S. Titi, An approximate inertial manifold approach to postprocessing Galerkin methods for the Navier-Stokes equations, Math. Comp., 68 (1999), 893-911.  doi: 10.1090/S0025-5718-99-01057-1.

[12]

B. García-Archilla and E. S. Titi, Postprocessing the Galerkin method: the finite-element case, SIAM J. Numer. Anal., 37 (2000), 470-499.  doi: 10.1137/S0036142998335893.

[13]

V. Girault and J.-L. Lions, Two-grid finite element scheme for the transient Navier-Stokes problem, M2AN Math. Model. Numer. Anal., 35 (2001), 945-980.  doi: 10.1051/m2an:2001145.

[14]

V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, Berlin Heidelberg, 1986. doi: 10.1007/978-3-642-61623-5.

[15]

R. Glowinski, Finite Element Methods for Incompressible Viscous Flow, Handbook of numerical analysis, Vol. IX, 3–1176, Handb. Numer. Anal., IX, North-Holland, Amsterdam, 2003.

[16]

J.-L. Guermond, Stabilization of Galerkin approximations of transport equations by subgrid modeling, M2AN Math. Model. Numer. Anal., 33 (1999), 1293-1316.  doi: 10.1051/m2an:1999145.

[17]

J.-L. Guermond, Subgrid stabilization of Galerkin approximations of linear contraction semi-groups of class $C^0$ in Hilbert spaces, Numer. Meth. PDEs., 17 (2001), 1-25. 

[18]

J.-L. Guermond, A. Marra and L. Quartapelle, Subgrid stabilized projection method for 2D unsteady flows at high Reynolds numbers, Comput. Meth. Appl. Mech. Engrg., 195)(2006), 5857–5876. doi: 10.1016/j.cma.2005.08.016.

[19]

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.

[20]

Y. N. He, A two-level finite element Galerkin method for the nonstationary Navier-Stokes equations, I: Spatial discretization, J. Comput. Math., 22 (2004), 21-32. 

[21]

Y. N. He and K. M. Liu, A multi-level finite element method in space-time for the Navier-Stokes equations, Numer. Meth. PDEs., 21 (2005), 1052-1078. 

[22]

Y. N. HeH. L. Miao and C. F. Ren, A two-level finite element Galerkin method for the nonstationary Navier-Stokes equations, Ⅱ: Time discretization, J. Comput. Math., 22 (2004), 33-54. 

[23]

Y. N. He and W. 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.

[24]

F. Hecht, New development in Freefem++, J. Numer. Math., 20 (2012), 251-266.  doi: 10.1515/jnum-2012-0013.

[25]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981.

[26]

T. J. R. Hughes, Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid-scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Engrg., 127 (1995), 387-401.  doi: 10.1016/0045-7825(95)00844-9.

[27]

T. J. R. HughesL. Mazzei and K. E. Jansen, Large eddy simulation and the variational multiscale method, Comput. Vis. Sci., 3 (2000), 47-59.  doi: 10.1007/s007910050051.

[28]

V. John, Finite Element Methods for Incompressible Flow Problems, Springer Series in Computational Mathematics, 51. Springer, Cham, 2016. doi: 10.1007/978-3-319-45750-5.

[29]

A. Labovschii, A defect correction method for the time-dependent Navier-Stokes equations, Numer. Meth. PDEs., 25 (2009), 1-25.  doi: 10.1002/num.20329.

[30]

W. Layton, A connection between subgrid scale eddy viscosity and mixed methods, Appl. Math. Comput., 133 (2002), 147-157.  doi: 10.1016/S0096-3003(01)00228-4.

[31]

W. LaytonH. K. Lee and J. Peterson, A defect-correction method for the incompressible Navier-Stokes equations, Appl. Math. Comput., 129 (2002), 1-19.  doi: 10.1016/S0096-3003(01)00026-1.

[32]

M. A. Olshanskii, Two-level method and some a priori estimates in unsteady Navier-Stokes calculations, J. Comput. Appl. Math., 104 (1999), 173-191.  doi: 10.1016/S0377-0427(99)00056-4.

[33]

Y. Q. Shang, A two-level subgrid stabilized Oseen iterative method for the steady Navier-Stokes equations, J. Comput. Phys., 233 (2013), 210-226.  doi: 10.1016/j.jcp.2012.08.024.

[34]

J. Smagorinsky, General circulation experiments with the primitive equations, I: The basic experiments, Mon. Wea. Rev., 91 (1963), 99-164.  doi: 10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2.

[35]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam 1984.

[36]

K. Wang, A new defect correction method for the Navier-Stokes equations at high Reynolds numbers, Appl. Math. Comput., 216 (2010), 3252-3264.  doi: 10.1016/j.amc.2010.04.050.

[37]

Y. Zhang and Y. N. He, Assessment of subgrid-scale models for the incompressible Navier-Stokes equations, J. Comput. Appl. Math., 234 (2010), 593-604.  doi: 10.1016/j.cam.2009.12.051.

Figure 1.  The triangulation of the computational domain
Figure 2.  Temporal evolution of the drag coefficient (left), the lift coefficient (middle), and the difference of the pressure between the front and the back of the cylinder $ p(0.15,0.2)-p(0.25,0.2) $ (right)
Figure 3.  Computed $ u_{1} $-velocities for flow around a circular cylinder at $ T = 2, 4, 6, 7 $ and $ 8 $ (from top to bottom)
Figure 4.  Computed $ u_{2} $-velocities for flow around a circular cylinder at $ T = 2, 4, 6, 7 $ and $ 8 $ (from top to bottom)
Figure 5.  Computed isobars for flow around a circular cylinder at $ T = 2, 4, 6, 7 $ and $ 8 $ (from top to bottom)
Table 1.  Errors of the computed velocities in $ L^{2} $-norm
$ h $ $ \parallel u(T)-u_{h}^{N}\parallel_{0} $ rate $ \parallel u(T)-\widetilde{u}_{h}^{N}\parallel_{0} $ rate
$ 1/4 $ 0.00217733 - 0.000992868 -
$ 1/6 $ 0.000529178 3.48866 0.00016355 4.44792
$ 1/8 $ 0.000202786 3.33415 4.4057e-05 4.55932
$ 1/10 $ 9.95046e-05 3.19053 1.6059e-05 4.52272
$ 1/12 $ 5.64563e-05 3.10845 7.11378e-06 4.46594
$ 1/14 $ 3.52027e-05 3.06417 3.57486e-06 4.46387
$ 1/16 $ 2.34586e-05 3.03961 1.94755e-06 4.54840
$ h $ $ \parallel u(T)-u_{h}^{N}\parallel_{0} $ rate $ \parallel u(T)-\widetilde{u}_{h}^{N}\parallel_{0} $ rate
$ 1/4 $ 0.00217733 - 0.000992868 -
$ 1/6 $ 0.000529178 3.48866 0.00016355 4.44792
$ 1/8 $ 0.000202786 3.33415 4.4057e-05 4.55932
$ 1/10 $ 9.95046e-05 3.19053 1.6059e-05 4.52272
$ 1/12 $ 5.64563e-05 3.10845 7.11378e-06 4.46594
$ 1/14 $ 3.52027e-05 3.06417 3.57486e-06 4.46387
$ 1/16 $ 2.34586e-05 3.03961 1.94755e-06 4.54840
Table 2.  Errors of the computed velocities in $ H^{1} $-norm
$ h $ $ \parallel \nabla(u(T)-u_{h}^{N})\parallel_{0} $ rate $ \parallel \nabla(u(T)-\widetilde{u}_{h}^{N})\parallel_{0} $ rate
$ 1/4 $ 0.0508249 - 0.018681 -
$ 1/6 $ 0.0226069 1.99803 0.00493605 3.28251
$ 1/8 $ 0.0127961 1.97827 0.0018011 3.50444
$ 1/10 $ 0.00823732 1.97391 0.000813042 3.56439
$ 1/12 $ 0.00574502 1.97641 0.000424348 3.56638
$ 1/14 $ 0.00423382 1.98007 0.000245707 3.54467
$ 1/16 $ 0.00324871 1.9834 0.000153708 3.51292
$ h $ $ \parallel \nabla(u(T)-u_{h}^{N})\parallel_{0} $ rate $ \parallel \nabla(u(T)-\widetilde{u}_{h}^{N})\parallel_{0} $ rate
$ 1/4 $ 0.0508249 - 0.018681 -
$ 1/6 $ 0.0226069 1.99803 0.00493605 3.28251
$ 1/8 $ 0.0127961 1.97827 0.0018011 3.50444
$ 1/10 $ 0.00823732 1.97391 0.000813042 3.56439
$ 1/12 $ 0.00574502 1.97641 0.000424348 3.56638
$ 1/14 $ 0.00423382 1.98007 0.000245707 3.54467
$ 1/16 $ 0.00324871 1.9834 0.000153708 3.51292
Table 3.  Errors of the computed pressures in $ L^{2} $-norm
$ h $ $ \parallel p(T)-p_{h}^{N}\parallel_{0} $ rate $ \parallel p(T)-\widetilde{p}_{h}^{N})\parallel_{0} $ rate
$ 1/4 $ 0.16059 - 0.00173177 -
$ 1/6 $ 0.0713664 2.00024 0.00038025 3.7391
$ 1/8 $ 0.0401428 2.00007 0.000124192 3.88971
$ 1/10 $ 0.0256912 2.00003 5.17408e-05 3.92385
$ 1/12 $ 0.0178411 2.00001 2.55797e-05 3.86376
$ 1/14 $ 0.0131077 2 1.46068e-05 3.63484
$ 1/16 $ 0.0100356 2 9.62881e-06 3.12082
$ h $ $ \parallel p(T)-p_{h}^{N}\parallel_{0} $ rate $ \parallel p(T)-\widetilde{p}_{h}^{N})\parallel_{0} $ rate
$ 1/4 $ 0.16059 - 0.00173177 -
$ 1/6 $ 0.0713664 2.00024 0.00038025 3.7391
$ 1/8 $ 0.0401428 2.00007 0.000124192 3.88971
$ 1/10 $ 0.0256912 2.00003 5.17408e-05 3.92385
$ 1/12 $ 0.0178411 2.00001 2.55797e-05 3.86376
$ 1/14 $ 0.0131077 2 1.46068e-05 3.63484
$ 1/16 $ 0.0100356 2 9.62881e-06 3.12082
Table 4.  A comparison of computational time in seconds of the methods
$ h $ $ 1/4 $ $ 1/6 $ $ 1/8 $ $ 1/10 $ $ 1/12 $ $ 1/14 $ $ 1/16 $
S-FEM 7.8146 17.1435 30.0884 46.7738 69.0377 93.4098 122.664
SP-FEM 7.8334 17.1858 30.1654 46.8942 69.2128 93.6512 122.973
$ h $ $ 1/4 $ $ 1/6 $ $ 1/8 $ $ 1/10 $ $ 1/12 $ $ 1/14 $ $ 1/16 $
S-FEM 7.8146 17.1435 30.0884 46.7738 69.0377 93.4098 122.664
SP-FEM 7.8334 17.1858 30.1654 46.8942 69.2128 93.6512 122.973
Table 5.  A comparison of the computed solutions by differential methods with $ P_3-P_2 $ elements for the postprocessing
Method $ \nu $ $ \parallel u(T)-\widetilde{u}_{h}^{N}\parallel_{0} $ $ \parallel \nabla(u(T)-\widetilde{u}_{h}^{N})\|_0 $ $ \parallel p(T)-\widetilde{p}_{h}^{N})\parallel_{0} $ CPU
Present $ 1 $ 5.41056e-08 6.5595e-06 5.55593e-06 803.389
$ 10^{-1} $ 4.25516e-07 7.98607e-06 5.27302e-06 805.513
$ 10^{-2} $ 4.52353e-06 4.51348e-05 5.19797e-06 820.973
$ 10^{-3} $ 4.5248e-05 0.000441765 5.1803e-06 800.503
$ 10^{-4} $ 0.000398027 0.00350141 5.16923e-06 779.926
$ 10^{-5} $ 0.00179621 0.0145128 5.17307e-06 779.998
$ 10^{-6} \; \; \; $ 0.00276865 0.022324 5.1727e-06 783.388
Ref. [3] $ 1 $ 5.18142e-08 6.5516e-06 5.54746e-06 718.683
$ 10^{-1} $ 4.14903e-07 7.36316e-06 5.25698e-06 718.624
$ 10^{-2} $ 4.51539e-06 3.36283e-05 5.20106e-06 722.98
$ 10^{-3} $ 4.59266e-05 0.000333893 5.17257e-06 732.069
$ 10^{-4} $ 0.000460089 0.00334075 5.16772e-06 740.576
$ 10^{-5} $ 0.00460161 0.0334164 5.16846e-06 735.447
$ 10^{-6} \; \; \; $ 0.0460168 0.334242 3.85961e-05 738.079
Method $ \nu $ $ \parallel u(T)-\widetilde{u}_{h}^{N}\parallel_{0} $ $ \parallel \nabla(u(T)-\widetilde{u}_{h}^{N})\|_0 $ $ \parallel p(T)-\widetilde{p}_{h}^{N})\parallel_{0} $ CPU
Present $ 1 $ 5.41056e-08 6.5595e-06 5.55593e-06 803.389
$ 10^{-1} $ 4.25516e-07 7.98607e-06 5.27302e-06 805.513
$ 10^{-2} $ 4.52353e-06 4.51348e-05 5.19797e-06 820.973
$ 10^{-3} $ 4.5248e-05 0.000441765 5.1803e-06 800.503
$ 10^{-4} $ 0.000398027 0.00350141 5.16923e-06 779.926
$ 10^{-5} $ 0.00179621 0.0145128 5.17307e-06 779.998
$ 10^{-6} \; \; \; $ 0.00276865 0.022324 5.1727e-06 783.388
Ref. [3] $ 1 $ 5.18142e-08 6.5516e-06 5.54746e-06 718.683
$ 10^{-1} $ 4.14903e-07 7.36316e-06 5.25698e-06 718.624
$ 10^{-2} $ 4.51539e-06 3.36283e-05 5.20106e-06 722.98
$ 10^{-3} $ 4.59266e-05 0.000333893 5.17257e-06 732.069
$ 10^{-4} $ 0.000460089 0.00334075 5.16772e-06 740.576
$ 10^{-5} $ 0.00460161 0.0334164 5.16846e-06 735.447
$ 10^{-6} \; \; \; $ 0.0460168 0.334242 3.85961e-05 738.079
Table 6.  A comparison of the computed solutions by differential methods with $ P_{2}-P_{1} $ elements on a finer mesh for the postprocessing
Method $ \nu $ $ \parallel u(T)-\widetilde{u}_{\widetilde{h}}^{N}\parallel_{0} $ $ \parallel \nabla(u(T)-\widetilde{u}_{\widetilde{h}}^{N})\|_0 $ $ \parallel p(T)-\widetilde{p}_{\widetilde{h}}^{N})\parallel_{0} $ CPU
Present $ 1 $ 1.88993e-06 0.000148371 0.000266744 23.2019
$ 10^{-1} $ 1.80946e-05 0.000208457 0.000266742 23.161
$ 10^{-2} $ 0.00016378 0.00141076 0.000266724 23.188
$ 10^{-3} $ 0.00100617 0.00903462 0.000266679 23.2399
$ 10^{-4} $ 0.00208262 0.0218984 0.000266653 23.0764
$ 10^{-5} $ 0.00233199 0.0255756 0.000266624 23.7189
$ 10^{-6}\; \; \; $ 0.00236024 0.0260096 0.000266632 24.2022
Ref. [3] $ 1 $ 1.86059e-06 0.000148096 0.000266745 21.2937
$ 10^{-1} $ 1.86303e-05 0.000193416 0.000266733 21.2211
$ 10^{-2} $ 0.00018424 0.00133237 0.000266741 21.1333
$ 10^{-3} $ 0.00183899 0.0133119 0.000266752 21.1595
$ 10^{-4} $ 0.0183863 0.133176 0.000266773 21.1786
$ 10^{-5} $ 0.183859 1.3318 0.000267064 21.1574
$ 10^{-6} \; \; \; $ 1.83858 13.318 0.000272255 21.88
Method $ \nu $ $ \parallel u(T)-\widetilde{u}_{\widetilde{h}}^{N}\parallel_{0} $ $ \parallel \nabla(u(T)-\widetilde{u}_{\widetilde{h}}^{N})\|_0 $ $ \parallel p(T)-\widetilde{p}_{\widetilde{h}}^{N})\parallel_{0} $ CPU
Present $ 1 $ 1.88993e-06 0.000148371 0.000266744 23.2019
$ 10^{-1} $ 1.80946e-05 0.000208457 0.000266742 23.161
$ 10^{-2} $ 0.00016378 0.00141076 0.000266724 23.188
$ 10^{-3} $ 0.00100617 0.00903462 0.000266679 23.2399
$ 10^{-4} $ 0.00208262 0.0218984 0.000266653 23.0764
$ 10^{-5} $ 0.00233199 0.0255756 0.000266624 23.7189
$ 10^{-6}\; \; \; $ 0.00236024 0.0260096 0.000266632 24.2022
Ref. [3] $ 1 $ 1.86059e-06 0.000148096 0.000266745 21.2937
$ 10^{-1} $ 1.86303e-05 0.000193416 0.000266733 21.2211
$ 10^{-2} $ 0.00018424 0.00133237 0.000266741 21.1333
$ 10^{-3} $ 0.00183899 0.0133119 0.000266752 21.1595
$ 10^{-4} $ 0.0183863 0.133176 0.000266773 21.1786
$ 10^{-5} $ 0.183859 1.3318 0.000267064 21.1574
$ 10^{-6} \; \; \; $ 1.83858 13.318 0.000272255 21.88
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