Skip to main content
SpringerLink
Log in

A parallel stabilized finite element method based on the lowest equal-order elements for incompressible flows

  • Published:
Computing Aims and scope Submit manuscript

Abstract

Based on a fully overlapping domain decomposition technique, a parallel stabilized equal-order finite element method for the steady Stokes equations is presented and studied. In this method, each processor computes a local stabilized finite element solution in its own subdomain by solving a global problem on a global mesh that is locally refined around its subdomain, where the lowest equal-order finite element pairs (continuous piecewise linear, bilinear or trilinear velocity and pressure) are used for the finite element discretization and a pressure-projection-based stabilization method is employed to circumvent the discrete inf–sup condition that is invalid for the used finite element pairs. The parallel stabilized method is unconditionally stable, free of parameter and calculation of derivatives, and is easy to implement based on an existing sequential solver. Optimal error estimates are obtained by the theoretical tool of local a priori error estimates for finite element solutions. Numerical results are also given to verify the theoretical predictions and illustrate the effectiveness of the method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Bank RE, Holst M (2000) A new paradigm for parallel adaptive meshing algorithms. SIAM J Sci Comput 22:1411–1443

    Article  MathSciNet  Google Scholar 

  2. Bank RE, Jimack PK (2001) A new parallel domain decomposition method for the adaptive finite element solution of elliptic partial differential equations. Concurr Comput Pract Expert 13:327–350

    Article  Google Scholar 

  3. Barth T, Bochev P, Gunzburger M, Shahid J (2004) A taxonomy of consistently stabilized finite element methods for the Stokes problem. SIAM J Sci Comput 25(5):1585–1607

    Article  MathSciNet  Google Scholar 

  4. Becker R, Braack M (2001) A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38(4):173–199

    Article  MathSciNet  Google Scholar 

  5. Bochev PB, Dohrmann CR, Gunzburger MD (2006) Stabilization of low-order mixed finite elements for the Stokes equations. SIAM J Numer Anal 44(1):82–101

    Article  MathSciNet  Google Scholar 

  6. Bochev P, Gunzburger M (2004) An absolutely stable pressure-Poisson stabilized finite element method for the Stokes equations. SIAM J Numer Anal 42:1189–1207

    Article  MathSciNet  Google Scholar 

  7. Brezzi F, Fortin M (2001) A minimal stabilisation procedure for mixed finite element methods. Numer Math 89:457–491

    Article  MathSciNet  Google Scholar 

  8. Codina R, Blasco J (1997) A finite element formulation for the Stokes problem allowing equal velocity-pressure interpolation. Comput Methods Appl Mech Eng 143(3–4):373–391

    Article  MathSciNet  Google Scholar 

  9. Codina R, Blasco J (2000) Analysis of a pressure-stabilized finite element approximation of the stationary Navier–Stokes equations. Numer Math 87(1):59–81

    Article  MathSciNet  Google Scholar 

  10. Douglas J, Wang J (1989) An absolutely stabilized finite element method for the Stokes problem. Math Comput 52:495–508

    Article  MathSciNet  Google Scholar 

  11. Franca L, Hughes T (1993) Convergence analyses of Galerkin-least-squares methods for symmetric advective–diffusive forms of the Stokes and incompressible Navier–Stokes equations. Comput Methods Appl Mech Eng 105(2):285–298

    Article  MathSciNet  Google Scholar 

  12. Ganesan S, Matthies G, Tobiska L (2008) Local projection stabilization of equal order interpolation of the Stokes problem. Math Comput 77(264):2039–2060

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. He YN, Xu JC, Zhou AH, Li J (2008) Local and parallel finite element algorithms for the Stokes problem. Numer Math 109(3):415–434

    Article  MathSciNet  Google Scholar 

  15. He YN, Xu JC, Zhou AH (2006) Local and parallel finite element algorithms for the Navier–Stokes problem. J Comput Math 24(3):227–238

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Hughes T, Franca L, Balestra M (1986) A new finite element formulation for computational fluid dynamics: V. Circumventing the Babu\(\breve{s}\)ka–Brezzi condition: a stable Petrov–Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput Methods Appl Mech Eng 59(1):85–99

    Article  Google Scholar 

  18. Hughes TJR, Liu W, Brooks A (1979) Finite element analysis for incompressible viscous flows by penalty function formulation. J Comput Phys 30:1–60

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Mitchell WF (2004) Parallel adaptive multilevel methods with full domain partitions. Appl Numer Anal Comput Math 1(1–2):36–48

    Article  MathSciNet  Google Scholar 

  21. Shang YQ (2013) 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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Shang YQ, He YN (2010) Parallel finite element algorithm based on full domain partition for stationary Stokes equations. Appl Math Mech Engl Ed 31(5):643–650

    Article  MathSciNet  Google Scholar 

  24. Shang YQ, He YN (2010) Parallel iterative finite element algorithms based on full domain partition for the stationary Navier–Stokes equations. Appl Numer Math 60(7):719–737

    Article  MathSciNet  Google Scholar 

  25. Vey S, Voigt A (2007) Adaptive full domain covering meshes for parallel finite element computations. Computing 81(1):53–75

    Article  MathSciNet  Google Scholar 

  26. Xu JC, Zhou AH (2000) Local and parallel finite element algorithms based on two-grid discretizations. Math Comput 69:881–909

    Article  MathSciNet  Google Scholar 

  27. Xu JC, Zhou AH (2001) Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problems. Adv Comput Math 14:293–327

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Yueqiang Shang

Authors
  1. Yueqiang Shang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yueqiang Shang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by the Natural Science Foundation of China (No. 11361016), the Basic and Frontier Explore Program of Chongqing Municipality, China (Nos. cstc2016jcyjA0348, cstc2018jcyjAX0305), and Fundamental Research Funds for the Central Universities (No. XDJK2018B032).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00607-019-00729-0

Keywords

Mathematics Subject Classification

Search

Navigation