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doi: 10.3934/dcdsb.2022022
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Local and parallel finite element algorithms for the incompressible Navier-Stokes equations with damping

Eid Wassim and  Yueqiang Shang ,

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

* Corresponding author: Yueqiang Shang

Received  April 2021 Revised  December 2021 Early access February 2022

Fund Project: This work is supported by the Natural Science Foundation of Chongqing Municipality, China (No. cstc2021jcyj-msxmX1044)

Using two-grid discretizations strategy, we present some local and parallel finite element algorithms for simulating the steady incompressible Navier-Stokes equations with a nonlinear damping term. In these algorithms, we compute a solution of the Navier-Stokes system with a nonlinear damping term on a coarse grid, and then adjust the solution by some local and parallel procedures on overlapped fine grid subomains. With the use of theoretical tool of local a priori estimate of the finite element solution, we estimate the error bounds of the approximate solutions, and derive the algorithmic parameter scalings. Finally, we give some numerical results to verify the theoretical predictions and demonstrate the efficiency of the proposed algorithms.

Keywords: Navier-Stokes equations, nonlinear damping term, finite element method, parallel algorithm, two-grid method.
Mathematics Subject Classification: Primary: 65N30, 65N15, 65N55; Secondary: 76D05, 76M10.
Citation: Eid Wassim, Yueqiang Shang. Local and parallel finite element algorithms for the incompressible Navier-Stokes equations with damping. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022022
References:
[1]

S. S. Antman, The equations for large vibrations of strings, Amer. Math. Monthly, 87 (1980), 359-370.  doi: 10.1080/00029890.1980.11995034.

[2]

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223.  doi: 10.1007/s00220-003-0859-8.

[3]

D. Bresch, B. Desjardins and C. K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Comm. Partial Differential Equations, 28 (2003), 843-868.  doi: 10.1081/PDE-120020499.

[4]

X. Cai and Q. Jiu, Weak and strong solutions for the incompressible Navier–Stokes equations with damping, J. Math. Anal. Appl., 343 (2008), 799-809.  doi: 10.1016/j.jmaa.2008.01.041.

[5]

G. Du and L. Zuo, A parallel partition of unity scheme based on two-grid discretizations for the Navier–Stokes problem, J. Sci. Comput., 75 (2018), 1445-1462.  doi: 10.1007/s10915-017-0593-6.

[6]

Q. Du and M. Gunzburger, Analysis of a Ladyzhenskaya model for incompressible viscous flow, J. Math. Anal. Appl., 155 (1991), 21-45.  doi: 10.1016/0022-247X(91)90024-T.

[7]

Q. Du and M. Gunzburger, FE approximations of a Ladyzhenskaya model for stationary incompressible viscous flow, SIAM J. Numer. Anal., 27 (1990), 1-19.  doi: 10.1137/0727001.

[8]

V. Girault and P. A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, Lecture Notes in Mathematics, Berlin Springer Verlag, 1979.

[9]

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

[10]

V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109 (1994), 295-308.  doi: 10.1006/jdeq.1994.1051.

[11]

Y. He, A fully discrete stabilized finite-element method for the time-dependent Navier-Stokes problem, IMA J. Numer. Anal., 23 (2003), 665-691.  doi: 10.1093/imanum/23.4.665.

[12]

Y. He and J. Li, Convergence of three iterative methods based on the finite element discretization for the stationary Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg., 198 (2009), 1351-1359.  doi: 10.1016/j.cma.2008.12.001.

[13]

Y. He, L. Mei, Y. Shang and J. Cui, Newton iterative parallel finite element algorithm for the steady Navier-Stokes equations, J. Sci. Comput., 44 (2010), 92-106.  doi: 10.1007/s10915-010-9371-4.

[14]

Y. 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. 

[15]

Y. He, J. Xu and A. Zhou, Local and parallel finite element algorithms for the Navier-Stokes problem, J. Comput. Math., 24 (2006), 227-238. 

[16]

Y. He, J. Xu, A. Zhou and J. Li, Local and parallel finite element algorithms for the Stokes problem, Numer. Math., 109 (2008), 415-434.  doi: 10.1007/s00211-008-0141-2.

[17]

F. Hecht, New development in freefem++, J. Numer. Math., 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013.

[18]

J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal., 19 (1982), 275-311.  doi: 10.1137/0719018.

[19]

J. G. Heywood and R. Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem. Part IV: Error analysis for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), 353-384.  doi: 10.1137/0727022.

[20]

M. Li, D. Shi and Y. Dai, Stabilized low order finite elements for Stokes equations with damping, J. Math. Anal. Appl., 435 (2016), 646-660.  doi: 10.1016/j.jmaa.2015.10.040.

[21]

M. Li, D. Shi, Z. Li and H. Chen, Two-level mixed finite element methods for the Navier–Stokes equations with damping, J. Math. Anal. Appl., 470 (2019), 292-307.  doi: 10.1016/j.jmaa.2018.10.002.

[22]

D. Liu and K. Li, Finite element analysis of the Stokes equations with damping, Math. Numer. Sin., 32 (2010), 433-448. 

[23]

Y. Ping, H. Su and X. Feng, Parallel two-step finite element algorithm for the stationary incompressible magnetohydrodynamic equations, International Journal of Numerical Methods for Heat & Fluid Flow, 29 (2019), 2709-2727.  doi: 10.1108/HFF-10-2018-0552.

[24]

Y. Ping, H. Su, J. Zhao and X. Feng, Parallel two-step finite element algorithm based on fully overlapping domain decomposition for the time-dependent natural convection problem, International Journal of Numerical Methods for Heat & Fluid Flow, 30 (2019), 496-515.  doi: 10.1108/HFF-03-2019-0241.

[25]

H. Qiu, Y. Zhang and L. Mei, A Mixed-FEM for Navier–Stokes type variational inequality with nonlinear damping term, Comput. Math. Appl., 73 (2017), 2191-2207.  doi: 10.1016/j.camwa.2017.02.046.

[26]

H. Qiu, Y. Zhang, L. Mei and C. Xue, A penalty-FEM for Navier-Stokes type variational inequality with nonlinear damping term, Numer. Methods Partial Differential Equations, 33 (2017), 918-940.  doi: 10.1002/num.22130.

[27]

Y. Shang, A parallel stabilized finite element method based on the lowest equal-order elements for incompressible flows, Computing, 102 (2020), 65-81.  doi: 10.1007/s00607-019-00729-0.

[28]

Y. Shang and Y. He, Parallel iterative finite element algorithms based on full domain partition for the stationary Navier–Stokes equations, Appl. Numer. Math., 60 (2010), 719-737.  doi: 10.1016/j.apnum.2010.03.013.

[29]

Y. Shang and Y. He, A parallel Oseen-linearized algorithm for the stationary Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg., 209/212 (2012), 172-183.  doi: 10.1016/j.cma.2011.11.003.

[30]

Y. Shang and S. Huang, A parallel subgrid stabilized finite element method based on two-grid discretization for simulation of 2D/3D steady incompressible flows, J. Sci. Comput., 60 (2014), 564-583.  doi: 10.1007/s10915-013-9806-9.

[31]

Y. Shang and J. Qin, Parallel finite element variational multiscale algorithms for incompressible flow at high Reynolds numbers, Appl. Numer. Math., 117 (2017), 1-21.  doi: 10.1016/j.apnum.2017.01.018.

[32]

D. Shi and Z. Yu, Superclose and superconvergence of finite element discretizations for the Stokes equations with damping, Appl. Math. Comput., 219 (2013), 7693-7698.  doi: 10.1016/j.amc.2013.01.057.

[33]

Q. Tang and Y. Huang, Analysis of local and parallel algorithm for incompressible Magnetohydrodynamics flows by finite element iterative method, Commun. Comput. Phys., 25 (2019), 729-751.  doi: 10.4208/cicp.oa-2017-0153.

[34]

C. Taylor and P. Hood, A numerical solution of the Navier-Stokes equations using the finite element technique, Computers & Fluids, 1 (1973), 73-100.  doi: 10.1016/0045-7930(73)90027-3.

[35]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, 3$^{rd}$ edition, Studies in Mathematics and its Applications, 2. North-Holland Publishing Co., Amsterdam, 1984.

[36]

J. Yu, F. Shi and H. Zheng, Local and parallel finite element algorithms based on the partition of unity for the Stokes problem, SIAM J. Sci. Comput., 36 (2014), C547–C567. doi: 10.1137/130925748.

[37]

J. Xu and A. Zhou, Local and parallel finite element algorithms based on two-grid discretizations, Math. Comp., 69 (2000), 881-909.  doi: 10.1090/S0025-5718-99-01149-7.

[38]

J. Xu and A. Zhou, Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problems, Adv. Comput. Math., 14 (2001), 293-327.  doi: 10.1023/A:1012284322811.

[39]

G. Zhang, H. Su and X. Feng, A novel parallel two-step algorithm based on finite element discretization for the incompressible flow problem, Numerical Heat Transfer, Part B: Fundamentals, 73 (2018), 329-341.  doi: 10.1080/10407790.2018.1486647.

[40]

Z. Zhang, X. Wu and M. Lu, On the uniqueness of strong solution to the incompressible Navier–Stokes equations with damping, J. Math. Anal. Appl., 377 (2011), 414-419.  doi: 10.1016/j.jmaa.2010.11.019.

show all references

References:
[1]

S. S. Antman, The equations for large vibrations of strings, Amer. Math. Monthly, 87 (1980), 359-370.  doi: 10.1080/00029890.1980.11995034.

[2]

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223.  doi: 10.1007/s00220-003-0859-8.

[3]

D. Bresch, B. Desjardins and C. K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Comm. Partial Differential Equations, 28 (2003), 843-868.  doi: 10.1081/PDE-120020499.

[4]

X. Cai and Q. Jiu, Weak and strong solutions for the incompressible Navier–Stokes equations with damping, J. Math. Anal. Appl., 343 (2008), 799-809.  doi: 10.1016/j.jmaa.2008.01.041.

[5]

G. Du and L. Zuo, A parallel partition of unity scheme based on two-grid discretizations for the Navier–Stokes problem, J. Sci. Comput., 75 (2018), 1445-1462.  doi: 10.1007/s10915-017-0593-6.

[6]

Q. Du and M. Gunzburger, Analysis of a Ladyzhenskaya model for incompressible viscous flow, J. Math. Anal. Appl., 155 (1991), 21-45.  doi: 10.1016/0022-247X(91)90024-T.

[7]

Q. Du and M. Gunzburger, FE approximations of a Ladyzhenskaya model for stationary incompressible viscous flow, SIAM J. Numer. Anal., 27 (1990), 1-19.  doi: 10.1137/0727001.

[8]

V. Girault and P. A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, Lecture Notes in Mathematics, Berlin Springer Verlag, 1979.

[9]

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

[10]

V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109 (1994), 295-308.  doi: 10.1006/jdeq.1994.1051.

[11]

Y. He, A fully discrete stabilized finite-element method for the time-dependent Navier-Stokes problem, IMA J. Numer. Anal., 23 (2003), 665-691.  doi: 10.1093/imanum/23.4.665.

[12]

Y. He and J. Li, Convergence of three iterative methods based on the finite element discretization for the stationary Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg., 198 (2009), 1351-1359.  doi: 10.1016/j.cma.2008.12.001.

[13]

Y. He, L. Mei, Y. Shang and J. Cui, Newton iterative parallel finite element algorithm for the steady Navier-Stokes equations, J. Sci. Comput., 44 (2010), 92-106.  doi: 10.1007/s10915-010-9371-4.

[14]

Y. 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. 

[15]

Y. He, J. Xu and A. Zhou, Local and parallel finite element algorithms for the Navier-Stokes problem, J. Comput. Math., 24 (2006), 227-238. 

[16]

Y. He, J. Xu, A. Zhou and J. Li, Local and parallel finite element algorithms for the Stokes problem, Numer. Math., 109 (2008), 415-434.  doi: 10.1007/s00211-008-0141-2.

[17]

F. Hecht, New development in freefem++, J. Numer. Math., 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013.

[18]

J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal., 19 (1982), 275-311.  doi: 10.1137/0719018.

[19]

J. G. Heywood and R. Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem. Part IV: Error analysis for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), 353-384.  doi: 10.1137/0727022.

[20]

M. Li, D. Shi and Y. Dai, Stabilized low order finite elements for Stokes equations with damping, J. Math. Anal. Appl., 435 (2016), 646-660.  doi: 10.1016/j.jmaa.2015.10.040.

[21]

M. Li, D. Shi, Z. Li and H. Chen, Two-level mixed finite element methods for the Navier–Stokes equations with damping, J. Math. Anal. Appl., 470 (2019), 292-307.  doi: 10.1016/j.jmaa.2018.10.002.

[22]

D. Liu and K. Li, Finite element analysis of the Stokes equations with damping, Math. Numer. Sin., 32 (2010), 433-448. 

[23]

Y. Ping, H. Su and X. Feng, Parallel two-step finite element algorithm for the stationary incompressible magnetohydrodynamic equations, International Journal of Numerical Methods for Heat & Fluid Flow, 29 (2019), 2709-2727.  doi: 10.1108/HFF-10-2018-0552.

[24]

Y. Ping, H. Su, J. Zhao and X. Feng, Parallel two-step finite element algorithm based on fully overlapping domain decomposition for the time-dependent natural convection problem, International Journal of Numerical Methods for Heat & Fluid Flow, 30 (2019), 496-515.  doi: 10.1108/HFF-03-2019-0241.

[25]

H. Qiu, Y. Zhang and L. Mei, A Mixed-FEM for Navier–Stokes type variational inequality with nonlinear damping term, Comput. Math. Appl., 73 (2017), 2191-2207.  doi: 10.1016/j.camwa.2017.02.046.

[26]

H. Qiu, Y. Zhang, L. Mei and C. Xue, A penalty-FEM for Navier-Stokes type variational inequality with nonlinear damping term, Numer. Methods Partial Differential Equations, 33 (2017), 918-940.  doi: 10.1002/num.22130.

[27]

Y. Shang, A parallel stabilized finite element method based on the lowest equal-order elements for incompressible flows, Computing, 102 (2020), 65-81.  doi: 10.1007/s00607-019-00729-0.

[28]

Y. Shang and Y. He, Parallel iterative finite element algorithms based on full domain partition for the stationary Navier–Stokes equations, Appl. Numer. Math., 60 (2010), 719-737.  doi: 10.1016/j.apnum.2010.03.013.

[29]

Y. Shang and Y. He, A parallel Oseen-linearized algorithm for the stationary Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg., 209/212 (2012), 172-183.  doi: 10.1016/j.cma.2011.11.003.

[30]

Y. Shang and S. Huang, A parallel subgrid stabilized finite element method based on two-grid discretization for simulation of 2D/3D steady incompressible flows, J. Sci. Comput., 60 (2014), 564-583.  doi: 10.1007/s10915-013-9806-9.

[31]

Y. Shang and J. Qin, Parallel finite element variational multiscale algorithms for incompressible flow at high Reynolds numbers, Appl. Numer. Math., 117 (2017), 1-21.  doi: 10.1016/j.apnum.2017.01.018.

[32]

D. Shi and Z. Yu, Superclose and superconvergence of finite element discretizations for the Stokes equations with damping, Appl. Math. Comput., 219 (2013), 7693-7698.  doi: 10.1016/j.amc.2013.01.057.

[33]

Q. Tang and Y. Huang, Analysis of local and parallel algorithm for incompressible Magnetohydrodynamics flows by finite element iterative method, Commun. Comput. Phys., 25 (2019), 729-751.  doi: 10.4208/cicp.oa-2017-0153.

[34]

C. Taylor and P. Hood, A numerical solution of the Navier-Stokes equations using the finite element technique, Computers & Fluids, 1 (1973), 73-100.  doi: 10.1016/0045-7930(73)90027-3.

[35]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, 3$^{rd}$ edition, Studies in Mathematics and its Applications, 2. North-Holland Publishing Co., Amsterdam, 1984.

[36]

J. Yu, F. Shi and H. Zheng, Local and parallel finite element algorithms based on the partition of unity for the Stokes problem, SIAM J. Sci. Comput., 36 (2014), C547–C567. doi: 10.1137/130925748.

[37]

J. Xu and A. Zhou, Local and parallel finite element algorithms based on two-grid discretizations, Math. Comp., 69 (2000), 881-909.  doi: 10.1090/S0025-5718-99-01149-7.

[38]

J. Xu and A. Zhou, Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problems, Adv. Comput. Math., 14 (2001), 293-327.  doi: 10.1023/A:1012284322811.

[39]

G. Zhang, H. Su and X. Feng, A novel parallel two-step algorithm based on finite element discretization for the incompressible flow problem, Numerical Heat Transfer, Part B: Fundamentals, 73 (2018), 329-341.  doi: 10.1080/10407790.2018.1486647.

[40]

Z. Zhang, X. Wu and M. Lu, On the uniqueness of strong solution to the incompressible Navier–Stokes equations with damping, J. Math. Anal. Appl., 377 (2011), 414-419.  doi: 10.1016/j.jmaa.2010.11.019.

Figure 1.  A comparison of the computed velocities with different numbers of subdomains, different damping parameters $ \alpha $ and $ r $
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Figure 2.  Comparison of $ u_1 $-velocity profile along the vertical centerline for the lid driven cavity flow at $ \nu = 0.01 $
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Figure 3.  Comparison of $ u_2 $-velocity profile along the horizontal centerline for the lid driven cavity flow at $ \nu = 0.01 $
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Figure 4.  A comparison of computed streamlines by Algorithm A1 with the reference data for lid-driven cavity flow: $ \nu = 0.01, \alpha = 1, r = 3 $
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Figure 5.  A comparison of computed streamlines by Algorithm A1 with the reference data for lid-driven cavity flow: $ \nu = 0.01, \alpha = 10, r = 6 $
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Figure 6.  Computed streamlines by Algorithm A1 with $ 4\times 4 $ sub-domains for lid-driven cavity flow at $ \nu = 0.01 $
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Figure 7.  Computed streamlines by Algorithm A1 for lid-driven cavity flow with $ 2\times 2 $ subdomains, $ \nu = 1 $ and $ r = 3 $
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Figure 8.  Computed streamlines by Algorithm A1 for lid-driven cavity flow with $ 4\times 4 $ subdomains, $ \nu = 1 $ and $ r = 3 $
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Table 1.  Errors of the approximate solutions obtained by present parallel finite element algorithm (Algorithm A1)
$ h $ $ H $ $ it $ CPU(s) $ |\lVert \nabla u -\nabla u_h |\rVert $$ _{0, \Omega} $ $ |\lVert p - p_h |\rVert $$ _{0, \Omega} $ Rate
1/27 1/18 3 0.923 0.000108646 0.000341526 -
1/64 1/32 3 2.634 2.06697e-05 6.35083e-05 1.94277
1/125 1/50 3 9.141 5.43566e-06 1.63504e-05 2.01912
1/216 1/72 3 22.593 1.97845e-06 5.53918e-06 1.94531
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$ h $ $ H $ $ it $ CPU(s) $ |\lVert \nabla u -\nabla u_h |\rVert $$ _{0, \Omega} $ $ |\lVert p - p_h |\rVert $$ _{0, \Omega} $ Rate
1/27 1/18 3 0.923 0.000108646 0.000341526 -
1/64 1/32 3 2.634 2.06697e-05 6.35083e-05 1.94277
1/125 1/50 3 9.141 5.43566e-06 1.63504e-05 2.01912
1/216 1/72 3 22.593 1.97845e-06 5.53918e-06 1.94531
Table 2.  Errors of the approximate solutions by the standard two-level finite element method
$ h $ $ H $ $ it $ CPU(s) $ \lVert \nabla u -\nabla u_h \rVert $$ _{0, \Omega} $ $ \lVert p - p_h \rVert $$ _{0, \Omega} $ Rate
1/27 1/18 2 0.801 0.000115269 0.000354182 -
1/64 1/32 2 4.255 2.05843e-05 6.30369e-05 1.99904
1/125 1/50 2 15.352 5.43494e-06 1.65252e-05 1.99732
1/216 1/72 2 36.999 1.92045e-06 5.53561e-06 1.9749
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$ h $ $ H $ $ it $ CPU(s) $ \lVert \nabla u -\nabla u_h \rVert $$ _{0, \Omega} $ $ \lVert p - p_h \rVert $$ _{0, \Omega} $ Rate
1/27 1/18 2 0.801 0.000115269 0.000354182 -
1/64 1/32 2 4.255 2.05843e-05 6.30369e-05 1.99904
1/125 1/50 2 15.352 5.43494e-06 1.65252e-05 1.99732
1/216 1/72 2 36.999 1.92045e-06 5.53561e-06 1.9749
Table 3.  A comparision of the methods with different values of viscosity
Method $ \nu $ $ it $ CPU(s) $ |\lVert\nabla u -\nabla u_h |\rVert $$ _{0, \Omega} $ $ |\lVert p - p_h |\rVert $$ _{0, \Omega} $
Present 1 2 2.251 2.06697e-05 6.30688e-05
0.1 3 2.959 2.06697e-05 6.35083e-05
0.01 4 3.438 2.06698e-05 9.93789e-05
0.001 8 5.852 2.06818e-05 0.000770386
Two-level method 1 2 4.025 2.05843e-05 6.30492e-05
0.1 2 4.065 2.05843e-05 6.30369e-05
0.01 3 4.681 2.05845e-05 6.30368e-05
0.001 6 6.229 2.05969e-05 6.30368e-05
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Method $ \nu $ $ it $ CPU(s) $ |\lVert\nabla u -\nabla u_h |\rVert $$ _{0, \Omega} $ $ |\lVert p - p_h |\rVert $$ _{0, \Omega} $
Present 1 2 2.251 2.06697e-05 6.30688e-05
0.1 3 2.959 2.06697e-05 6.35083e-05
0.01 4 3.438 2.06698e-05 9.93789e-05
0.001 8 5.852 2.06818e-05 0.000770386
Two-level method 1 2 4.025 2.05843e-05 6.30492e-05
0.1 2 4.065 2.05843e-05 6.30369e-05
0.01 3 4.681 2.05845e-05 6.30368e-05
0.001 6 6.229 2.05969e-05 6.30368e-05
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