American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2020222

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

Yueqiang Shang , and  Qihui Zhang

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

* Corresponding author: Yueqiang Shang

Received  January 2020 Revised  May 2020 Published  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.

Keywords: Navier-Stokes equations, postprocessing, subgrid stabilization, mixed finite element method.
Mathematics Subject Classification: Primary: 35Q30, 65M15, 65M60; Secondary: 76D05.
Citation: Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020222
References:
[1]

H. Abboud, V. 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.  Google Scholar

[2]

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

[3]

B. Ayuso, B. 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.  Google Scholar

[4]

B. Ayuso, J. 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.  Google Scholar

[5]

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

[6]

J. de Frutos, B. García-Archilla, V. 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.  Google Scholar

[7]

J. de Frutos, B. 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.  Google Scholar

[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.  Google Scholar

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

B. García-Archilla, J. 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.  Google Scholar

[11]

B. García-Archilla, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[22]

Y. N. He, H. 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.   Google Scholar

[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.  Google Scholar

[24]

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

[25]

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

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

W. Layton, H. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

H. Abboud, V. 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.  Google Scholar

[2]

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

[3]

B. Ayuso, B. 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.  Google Scholar

[4]

B. Ayuso, J. 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.  Google Scholar

[5]

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

[6]

J. de Frutos, B. García-Archilla, V. 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.  Google Scholar

[7]

J. de Frutos, B. 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.  Google Scholar

[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.  Google Scholar

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

B. García-Archilla, J. 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.  Google Scholar

[11]

B. García-Archilla, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[22]

Y. N. He, H. 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.   Google Scholar

[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.  Google Scholar

[24]

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

[25]

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

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

W. Layton, H. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  The triangulation of the computational domain
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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)
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Figure 3.  Computed $ u_{1} $-velocities for flow around a circular cylinder at $ T = 2, 4, 6, 7 $ and $ 8 $ (from top to bottom)
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Figure 4.  Computed $ u_{2} $-velocities for flow around a circular cylinder at $ T = 2, 4, 6, 7 $ and $ 8 $ (from top to bottom)
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Figure 5.  Computed isobars for flow around a circular cylinder at $ T = 2, 4, 6, 7 $ and $ 8 $ (from top to bottom)
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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
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$ 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
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$ 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
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$ 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
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$ 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
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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
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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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