A parallel stabilized quadratic equal-order finite element algorithm for the steady Navier–Stokes equations

References

• R. Adams, Sobolev Spaces, Academaic Press Inc., New York, 1975.
   Google Scholar
• P. Bochev, C.R. Dohrmann, and M.D. Gunzburger, Stabilization of low-order mixed finite elements for the Stokes equations, SIAM J. Numer. Anal. 44 (2006), pp. 82–101.
   Web of Science ®Google Scholar
• Y.Z. Cao, M. Gunzburger, and X.M. He, Parallel, non-iterative, multi-physics domain decomposition methods for time-dependent Stokes–Darcy systems, Math. Comput. 83 (2014), pp. 1617–1644.
   Web of Science ®Google Scholar
• X.J. Dong, Y.N. He, H.B. Wei, and Y.H. Zhang, Local and parallel finite element algorithm based on the partition of unity method for the incompressible MHD flow, Adv. Comput. Math. 44 (2018), pp. 1295–1319.
   Web of Science ®Google Scholar
• G.Z. Du and L.Y. Zuo, A parallel partition of unity scheme based on two-grid discretizations for the Navier–Stokes problem, J. Sci. Comput. 4 (2017), pp. 1–18.
   Google Scholar
• G.Z. Du and L.Y. Zuo, Local and parallel finite element methods for the coupled Stokes/Darcy model, Numer. Algor. doi:10.1007/s11075-020-01021-5
   Web of Science ®Google Scholar
• D.K. Gartling, A test problem for outflow boundary conditions-flow over a backward-facing step, Int. J. Numer. Methods Fluids 11 (1990), pp. 953–967.
   Web of Science ®Google Scholar
• U. Ghia, K.N. Ghia, and C.T. Shin, High-Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method, J. Comput. Phys. 48 (1982), pp. 387–441.
   Web of Science ®Google Scholar
• V. Girault and P.A. Raviart, Finite Element Method for Navier–Stokes Equations: Theory and Algorithms, Springer-Verlag, Berlin, Heidelberg, 1986.
   Google Scholar
• Y.N. He and J. Li, A stabilized finite element method based on local polynomial pressure projection for the stationary Navier–Stokes equations, Appl. Numer. Math. 58 (2008), pp. 1503–1514.
   Web of Science ®Google Scholar
• Y.N. 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. Eng. 198 (2009), pp. 1351–1359.
   Web of Science ®Google Scholar
• Y.N. He, J.C. Xu, and A.H. Zhou, Local and parallel finite element algorithms for the Navier–Stokes problem, J. Comput. Math. 24 (2006), pp. 227–238.
   Web of Science ®Google Scholar
• Y.N. He, J.C. Xu, A.H. Zhou, and J. Li, Local and parallel finite element algorithms for the Stokes problem, Numer. Math. 109 (2008), pp. 415–434.
   Web of Science ®Google Scholar
• Y.N. He, L.Q. Mei, Y.Q. Shang, and J. Cui, Newton iterative parallel finite element algorithm for the steady Navier–Stokes equations, J. Sci. Comput. 44 (2010), pp. 92–106.
   Web of Science ®Google Scholar
• F. Hecht, New development in FreeFem++, J. Numer. Math. 20 (2012), pp. 251–265.
   Web of Science ®Google Scholar
• P.Z. Huang, X.L. Feng, and D.M. Liu, Two-level stabilized method based on three corrections for the stationary Navier–Stokes equations, Appl. Numer. Math. 62 (2012), pp. 988–1001.
   Web of Science ®Google Scholar
• P.Z. Hunag, X.L. Feng, and Y.N. He, An efficient two-step algorithm for the incompressible flow problem, Adv. Comput. Math. 41 (2015), pp. 1059–1077.
   Web of Science ®Google Scholar
• N. Kechkar and D. Silvester, Analysis of locally stabilized mixed finite element methods for the Stokes problem, Math. Comput. 58 (1992), pp. 1–10.
   Web of Science ®Google Scholar
• W. Layton, A connection between subgrid scale eddy viscosity and mixed methods, Appl. Math. Comput. 133 (2002), pp. 147–157.
   Web of Science ®Google Scholar
• Q.T. Li and G.Z. Du, Local and parallel finite element methods based on two-grid discretizations for the nonstationary Navier–Stokes equations, Numer. Algor. doi:10.1007/s11075-021-01100-1
   Web of Science ®Google Scholar
• J. Li and Y.N. He, A stabilized finite element method based on two local Gauss integrations for the Stokes equations, J. Comput. Appl. Math. 214 (2008), pp. 58–65.
   Web of Science ®Google Scholar
• Y. Ping, H.Y. Su, and X.L. Feng, Parallel two-step finite element algorithm for the stationary incompressible magnetohydrodynamic equations, Int. J. Numer. Methods Heat Fluid Flow. 29 (2019), pp. 2709–2727.
   Web of Science ®Google Scholar
• Y. Ping, H.Y. Su, J.P. Zhao, and X.L. Feng, Parallel two-step finite element algorithm based on fully overlapping domain decomposition for the time-dependent natural convection problem, Int. J. Numer. Methods Heat Fluid Flow. 30 (2020), pp. 496–515.
   Web of Science ®Google Scholar
• H.L. Qiu, R. An, L.Q. Mei, and C.F. Xue, Two-step algorithms for the stationary incompressible Navier–Stokes equations with friction boundary conditions, Appl. Numer. Math. 120 (2017), pp. 97–114.
   Web of Science ®Google Scholar
• Y.Q. Shang, A parallel subgrid stabilized finite element method based on fully overlapping domain decomposition for the Navier–Stokes equations, J. Math. Anal. Appl. 403 (2013), pp. 667–679.
   Web of Science ®Google Scholar
• Y.Q. Shang, A parallel finite element variational multiscale method based on fully overlapping domain decomposition for incompressible flows, Numer. Methods Partial Differ. Equ. 31 (2015), pp. 856–875.
   Web of Science ®Google Scholar
• Y.Q. Shang and Y.N. He, Parallel finite element algorithm based on full domain partition for the stationary Stokes equations, Appl. Math. Mech. (Engl. Ed.) 35 (2010), pp. 643–650.
   Google Scholar
• Y.Q. Shang and Y.N. He, Parallel iterative finite element algorithms based on fully domain decomposition for the stationary Navier–Stokes equations, Appl. Numer. Math. 60 (2010), pp. 719–737.
   Web of Science ®Google Scholar
• Y.Q. Shang and J. Qin, Parallel finite element variational multiscale algorithms for incompressible flow at high Reynolds numbers, Appl. Numer. Math. 117 (2017), pp. 1–21.
   Web of Science ®Google Scholar
• Y.Q. Shang and K. Wang, Local and parallel finite element algorithms based on two-grid discretizations for the transient Stokes equations, Numer. Algor. 54 (2010), pp. 195–218.
   Web of Science ®Google Scholar
• Y.Q. Shang, Y.N. He, and D.W. Kim, A new parallel finite element algorithm for the stationary Navier–Stokes equations, Finite Elem. Anal. Des. 47 (2011), pp. 1262–1279.
   Web of Science ®Google Scholar
• Y.Z. Sun, W.W. Sun, and H.B. Zheng, Domain decomposition method for the fully-mixed Stokes–Darcy coupled problem, Comput. Methods Appl. Mech. Eng. 374 (2021), p. 113578.
   Google Scholar
• R. Temam, Navier–Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam, 1984.
   Google Scholar
• Z.F. Weng, Z.J. Yang, and X.L. Lu, Two-level quadratic equal-order stabilized method for the Stokes eigenvalue problem, Int. J. Comput. Math. 92 (2015), pp. 337–348.
   Web of Science ®Google Scholar
• J.C. Xu and A.H. Zhou, Local and parallel finite element algorithms based on two-grid discretizations, Math. Comput. 69 (2000), pp. 881–909.
   Web of Science ®Google Scholar
• J.C. Xu and A.H. Zhou, Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problems, Adv. Comput. Math. 14 (2001), pp. 293–327.
   Web of Science ®Google Scholar
• J.P. Yu, H.B. Zheng, and F. Shi, Local and parallel finite element algorithms based on the partition of unity for the Stokes problem, SIAM J. Sci. Comput. 36 (2014), pp. C547–C567.
   Web of Science ®Google Scholar
• G.L. Zhang, H.Y. Su, and X.L. Feng, A novel parallel two-step algorithm based on finite element discretization for the incompressible flow problem, Numer. Heat Transf. B Fund. 73 (2018), pp. 329–341.
   Web of Science ®Google Scholar
• B. Zheng and Y.Q. Shang, Parallel iterative stabilized finite element algorithms based on the lowest equal-order elements for the stationary Navier–Stokes equations, Appl. Math. Comput. 357 (2019), pp. 35–56.
   Web of Science ®Google Scholar
• B. Zheng and Y.Q. Shang, A parallel stabilized finite element variational multiscale method based on fully overlapping domain decomposition for the incompressible Navier–Stokes equations, Appl. Numer. Math. 159 (2021), pp. 138–158.
   Web of Science ®Google Scholar
• B. Zheng and Y.Q. Shang, Local and parallel stabilized finite element methods based on full domain decomposition for the stationary Stokes equations, Int. J. Nonlinear Sci. Numer. Simul.doi:10.1515/ijnsns-2019-0191
   Web of Science ®Google Scholar
• H.B. Zheng, S. Li, and Y.R. Hou, A quadratic equal-order stabilized method for Stokes problem based on two local Gauss integrations, Numer. Methods Partial Differ. Equ. 26 (2010), pp. 1180–1190.
   Web of Science ®Google Scholar
• H.B. Zheng, J.P. Yu, and F. Shi, local and parallel finite element algorithm based on the partition of unity for incompressible flows, J. Sci. Comput. 65 (2015), pp. 512–532.
   Web of Science ®Google Scholar
• K.R. Zhou and Y.Q. Shang, Parallel iterative stabilized finite element algorithms for the Navier–Stokes equations with nonlinear slip boundary conditions, Int. J. Numer. Methods Fluids. 93 (2021), pp. 1074–1109.
   Web of Science ®Google Scholar
• L.Y. Zuo and G.Z. Du, A parallel two-grid linearized method for the coupled Navier–Stokes–Darcy problem, Numer. Algor. 77 (2018), pp. 151–165.
   Web of Science ®Google Scholar