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Parallel iterative stabilized finite element methods based on the quadratic equal-order elements for incompressible flows

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Abstract

Combining the quadratic equal-order stabilized method with the approach of local and parallel finite element computations and classical iterative methods for the discretization of the steady-state Navier–Stokes equations, three parallel iterative stabilized finite element methods based on fully overlapping domain decomposition are proposed and compared in this paper. In these methods, each processor independently computes an approximate solution in its own subdomain using a global composite mesh that is fine around its own subdomain and coarse elsewhere, making the methods be easy to implement based on existing codes and have low communication complexity. Under some (strong) uniqueness conditions, stability and convergence theory of the parallel iterative stabilized methods are derived. Numerical tests are also performed to demonstrate the stability, convergence orders and high efficiency of the proposed methods.

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References

  1. Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis. North-Holland, Amsterdam (1984)

    MATH  Google Scholar 

  2. Girault, V., Raviart, P.A.: Finite Element Approximation of the Navier–Stokes Equations. Springer, Berlin (1979)

    Book  Google Scholar 

  3. Girault, V., Raviart, P.A.: Finite Element Method for Navier–Stokes Equations: Theory and Algorithms. Springer, Berlin (1987)

    MATH  Google Scholar 

  4. Hughes, T.J.R., Franca, L.P., Balestra, M.: A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuska–Brezzi condition: a stable Petrov–Galerkin formulation of the Stokes problem accomodating equal-order interpolations. Comput. Methods Appl. Mech. Eng. 59, 85–99 (1986)

    Article  Google Scholar 

  5. Brooks, A.N., Hughes, T.J.R.: Streamline upwind/Petrov–Galerkin formulations for convective dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 32, 199–259 (1982)

    Article  Google Scholar 

  6. Hughes, T.J.R., Franca, L.P., Hulbert, G.M.: A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective–diffusive equations. Comput. Methods Appl. Mech. Eng. 73, 173–189 (1989)

    Article  MathSciNet  Google Scholar 

  7. Masud, A., Khurram, R.A.: A multiscale finite element method for the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 195, 1750–1777 (2006)

    Article  MathSciNet  Google Scholar 

  8. Blasco, J., Codina, R.: Stabilization finite elements method for the transient Navier–Stokes equations based on a pressure gradient projection. Comput. Methods Appl. Mech. Eng. 182, 277–300 (2000)

    Article  Google Scholar 

  9. Li, J., He, Y.N.: A stabilized finite element method based on two local Gauss integrations for the Stokes equations. J. Comput. Appl. Math. 214, 58–65 (2008)

    Article  MathSciNet  Google Scholar 

  10. He, Y.N., Li, J.: A stabilized finite element method based on local polynomial pressure projection for the stationary Navier–Stokes equations. Appl. Numer. Math. 58, 1503–1514 (2008)

    Article  MathSciNet  Google Scholar 

  11. Zheng, H.B., Shan L., Hou, Y.R.: A quadratic equal-order stabilized method for Stokes problem based on two local Gauss integrations. Numer. Methods Partial Differ. Equ. 26, 180–1190 (2010)

    Article  MathSciNet  Google Scholar 

  12. Huang, P.Z., Feng, X.L., Liu, D.M.: Two-level stabilized method based on three corrections for the stationary Navier–Stokes equations. Appl. Numer. Math. 62, 988–1001 (2012)

    Article  MathSciNet  Google Scholar 

  13. Qiu, H.L., An, R., Mei, L.Q., Xue, C.F.: Two-step algorithms for the stationary incompressible Navier–Stokes equations with friction boundary conditions. Appl. Numer. Math. 120, 97–114 (2017)

    Article  MathSciNet  Google Scholar 

  14. Zhang, G.L., Su, H.Y., Feng, X.L.: A novel parallel two-step algorithm based on finite element discretization for the incompressible flow problem. Numer. Heat Transf. B Fundam. 73, 329–341 (2018)

    Article  Google Scholar 

  15. Zheng, B., Shang, Y.Q.: A parallel quadratic equal-order stabilized finite element algorithm based on fully overlapping domain decomposition for the steady Navier–Stokes equations, Submitted for publication

  16. Zheng, B., Shang, Y.Q.: Parallel iterative stabilized finite element algorithms based on the lowest equal-order elements for the stationary Navier–Stokes equations. Appl. Math. Comput. 357, 35–56 (2019)

    MathSciNet  MATH  Google Scholar 

  17. Xu, J.C., Zhou, A.H.: Local and parallel finite element algorithms based on two-grid discretizations. Math. Comput. 69, 881–909 (2000)

    Article  MathSciNet  Google Scholar 

  18. Xu, J.C., Zhou, A.H.: Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problem. Adv. Comput. Math. 14, 293–327 (2001)

    Article  MathSciNet  Google Scholar 

  19. He, Y.N., Xu, J.C., Zhou, A.H., Li, J.: Local and parallel finite element algorithms for the Stokes problem. Numer. Math. 109, 415–434 (2008)

    Article  MathSciNet  Google Scholar 

  20. Shang, Y.Q., Wang, K.: Local and parallel finite element algorithms based on two-grid discretizations for the transient Stokes equations. Numer. Algorithms 54, 195–218 (2010)

    Article  MathSciNet  Google Scholar 

  21. He, Y.N., Xu, J.C., Zhou, A.H.: Local and parallel finite element algorithms for the Navier–Stokes problem. J. Comput. Math. 24, 227–238 (2006)

    MathSciNet  MATH  Google Scholar 

  22. Shang, Y.Q., He, Y.N., Kim, D.W, Zhou, X.J.: A new parallel finite element algorithm for the stationary Navier–Stokes equations. Finite Elem. Anal. Des. 47, 1262–1279 (2011)

    Article  MathSciNet  Google Scholar 

  23. Shang, Y.Q., He, Y.N.: A parallel Oseen-linearized algorithm for the stationary Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 209, 172–183 (2012)

    Article  MathSciNet  Google Scholar 

  24. Du, G.Z., Hou, Y.R., Zuo, L.Y.: Local and parallel finite element method for the mixed Navier–Stokes/Darcy model. Int. J. Comput. Math. 93, 1155–1172 (2015)

    Article  MathSciNet  Google Scholar 

  25. Zhang, Y.H., Hou, Y.R., Shan, L., Dong, X.J.: Local and parallel finite element algorithm for stationary incompressible magnetohydrodynamics. Numer. Methods Partial Differ. Equ. 33, 1513–1539 (2017)

    Article  MathSciNet  Google Scholar 

  26. Tang, Q.L., Huang, Y.Q.: Local and parallel finite flement algorithm based on Oseen-type iteration for the stationary incompressible MHD flow. J. Sci. Comput. 70, 1–26 (2017)

    Article  MathSciNet  Google Scholar 

  27. Yang, Y.D., Han, J.Y.: The multilevel mixed finite element discretizations based on local defect-correction for the Stokes eigenvalue problem. Comput. Meth. Appl. Mech. Eng. 289, 249–266 (2015)

    Article  MathSciNet  Google Scholar 

  28. Bi, H., Han, J.Y., Yang, Y.D.: Local and parallel finite element algorithms for the transmission eigenvalue problem. J. Sci. Comput. 78, 351–375 (2019)

    Article  MathSciNet  Google Scholar 

  29. Zheng, H.B., Shi, F., Hou, Y.R., et al.: New local and parallel finite element algorithm based on the partition of unity. J. Math. Anal. Appl. 435, 1–19 (2016)

    Article  MathSciNet  Google Scholar 

  30. Du, G.Z., Zuo, L.Y.: A parallel partition of unity scheme based on two-grid discretizations for the Navier–Stokes problem. J. Sci. Comput. 4, 1–18 (2017)

    Google Scholar 

  31. Shang, Y.Q.: A parallel subgrid stabilized finite element method based on fully overlapping domain decomposition for the Navier–Stokes equations. J. Math. Anal. Appl. 403, 667–679 (2013)

    Article  MathSciNet  Google Scholar 

  32. Shang, Y.Q.: Parallel defect-correction algorithms based on finite element discretization for the Navier–Stokes equations. Comput. Fluids 79, 200–212 (2013)

    Article  MathSciNet  Google Scholar 

  33. Shang, Y.Q., Qin, J.: Parallel finite element variational multiscale algorithms for incompressible flow at high Reyonds numbers. Appl. Numer. Math. 117, 1–21 (2017)

    Article  MathSciNet  Google Scholar 

  34. He, Y.N., Li, J.: Convergence of three iterative methods based on the finite element discretization for the stationary Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 198, 1351–1359 (2009)

    Article  MathSciNet  Google Scholar 

  35. Shang, Y.Q., He, Y.N.: Parallel iterative finite element algorithms based on full domain decomposition for the stationary Navier–Stokes equations. Appl. Numer. Math. 60, 719–737 (2010)

    Article  MathSciNet  Google Scholar 

  36. Adams, R.: Sobolev Spaces. Academaic Press Inc, New York (1975)

    MATH  Google Scholar 

  37. Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier–Stokes problem. Part I: regularity of solutions and second-order spatial discretization. SIAM J. Numer. Anal. 19, 275–311 (1982)

    Article  MathSciNet  Google Scholar 

  38. Kechkar, N., Silvester, D.: Analysis of locally stabilized mixed finite element methods for the Stokes problem. Math. Comput. 58, 1–10 (1992)

    Article  MathSciNet  Google Scholar 

  39. Hecht, F.: New development in FreeFem++. J. Numer. Math. 20(3–4), 251–265 (2012)

    MathSciNet  MATH  Google Scholar 

  40. Arioli, M., Loghin, D., Wathen, A.J.: Stopping criteria for iterations in finite element methods. Numer. Math. 99(3), 381–410 (2005)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was 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). The authors appreciate the valuable comments and suggestions made by the reviewers which led to an improvement of the paper.

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  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China

    Bo Zheng & Yueqiang Shang

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  1. Bo Zheng
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  2. Yueqiang Shang
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Correspondence to Yueqiang Shang.

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Zheng, B., Shang, Y. Parallel iterative stabilized finite element methods based on the quadratic equal-order elements for incompressible flows. Calcolo 57, 34 (2020). https://doi.org/10.1007/s10092-020-00382-6

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