Skip to main content
SpringerLink
Log in

Analysis of a class of globally divergence-free HDG methods for stationary Navier-Stokes equations

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we analyze a class of globally divergence-free (and therefore pressure-robust) hybridizable discontinuous Galerkin (HDG) finite element methods for stationary Navier-Stokes equations. The methods use the \(\cal{P}_{k}/\cal{P}_{k-1}(k\geqslant 1)\) discontinuous finite element combination for the velocity and pressure approximations in the interior of elements, piecewise \(\cal{P}_{m}(m=k,k-1)\) for the velocity gradient approximation in the interior of elements, and piecewise \(\cal{P}_{k}/\cal{P}_{k}\) for the trace approximations of the velocity and pressure on the inter-element boundaries. We show that the uniqueness condition for the discrete solution is guaranteed by that for the continuous solution together with a sufficiently small mesh size. Based on the derived discrete HDG Sobolev embedding properties, optimal error estimates are obtained. Numerical experiments are performed to verify the theoretical analysis.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnold D N, Brezzi F, Cockburn B, et al. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J Numer Anal, 2002, 39: 1749–1779

    Article  MathSciNet  Google Scholar 

  2. Arnold D N, Brezzi F, Fortin M. A stable finite element for the Stokes equations. Calcolo, 1984, 21: 337–344

    Article  MathSciNet  Google Scholar 

  3. Auricchio F, da Veiga L B, Lovadina C, et al. The importance of the exact satisfaction of the incompressibility constraint in nonlinear elasticity: Mixed FEMs versus NURBS-based approximations. Comput Methods Appl Mech Engrg, 2010, 199: 314–323

    Article  MathSciNet  Google Scholar 

  4. Babuška I. The finite element method with Lagrangian multipliers. Numer Math, 1972, 20: 179–192

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Bochev P, Dohrmann C R, Gunzburger M D. Stabilization of low-order mixed finite elements for the Stokes equations. SIAM J Numer Anal, 2006, 44: 82–101

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Boffi D. The immersed boundary method for fluid-structure interactions: Mathematical formulation and numerical approximation. Boll Unione Mat Ital, 2012, 5: 711–724

    MathSciNet  Google Scholar 

  10. Boffi D, Brezzi F, Fortin M. Finite elements for the Stokes problem. In: Mixed Finite Elements, Compatibility Conditions, and Applications. Lecture Notes in Mathematics, vol. 1939. Berlin-Heidelberg: Springer, 2008, 45–100

    Chapter  Google Scholar 

  11. Brezzi F. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO Sér Rouge, 1974, 8: 129–151

    MathSciNet  Google Scholar 

  12. Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. New York: Springer-Verlag, 1991

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Burman E. Pressure projection stabilizations for Galerkin approximations of Stokes’ and Darcy’s problem. Numer Methods Partial Differential Equations, 2008, 24: 127–143

    Article  MathSciNet  Google Scholar 

  15. Cesmelioglu A, Cockburn B, Qiu W F. Analysis of a hybridizable discontinuous Galerkin method for the steady-state incompressible Navier-Stokes equations. Math Comp, 2017, 86: 1643–1670

    Article  MathSciNet  Google Scholar 

  16. Chen G, Feng M-F. A new absolutely stable simplified Galerkin least-squares finite element method using nonconforming element for the Stokes problem. Appl Math Comput, 2013, 219: 5356–5366

    MathSciNet  Google Scholar 

  17. Chen G, Feng M-F, He Y-N. Unified analysis for stabilized methods of low-order mixed finite elements for stationary Navier-Stokes equations. Appl Math Mech (English Ed), 2013, 34: 953–970

    Article  MathSciNet  Google Scholar 

  18. Chen G, Feng M-F, Xie X-P. Robust globally divergence-free weak Galerkin methods for Stokes equations. J Comput Math, 2016, 34: 549–572

    Article  MathSciNet  Google Scholar 

  19. Chen Y-M, Xie X-P. A streamline diffusion nonconforming finite element method for the time-dependent linearized Navier-Stokes equations. Appl Math Mech (English Ed), 2010, 31: 861–874

    Article  MathSciNet  Google Scholar 

  20. Cockburn B, Gopalakrishnan J. Incompressible finite elements via hybridization. Part I: The Stokes system in two space dimensions. SIAM J Numer Anal, 2005, 43: 1627–1650

    Article  MathSciNet  Google Scholar 

  21. Cockburn B, Gopalakrishnan J, Lazarov R. Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J Numer Anal, 2009, 47: 1319–1365

    Article  MathSciNet  Google Scholar 

  22. Cockburn B, Kanschat G, Schotzau D. A locally conservative LDG method for the incompressible Navier-Stokes equations. Math Comp, 2005, 74: 1067–1095

    Article  MathSciNet  Google Scholar 

  23. Cockburn B, Kanschat G, Schotzau D. A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations. J Sci Comput, 2007, 31: 61–73

    Article  MathSciNet  Google Scholar 

  24. Cockburn B, Kanschat G, Schötzau D. An equal-order DG method for the incompressible Navier-Stokes equations. J Sci Comput, 2009, 40: 188–210

    Article  MathSciNet  Google Scholar 

  25. Cockburn B, Karniadakis G E, Shu C-W. The development of discontinuous Galerkin methods. In: Discontinuous Galerkin Methods. Lecture Notes in Computational Science and Engineering, vol. 11. Berlin: Springer, 2000, 3–50

    Chapter  Google Scholar 

  26. Cockburn B, Nguyen N C, Peraire J. A comparison of HDG methods for Stokes flow. J Sci Comput, 2010, 45: 215–237

    Article  MathSciNet  Google Scholar 

  27. Cockburn B, Sayas F-J. Divergence-conforming HDG methods for Stokes flows. Math Comp, 2014, 83: 1571–1598

    Article  MathSciNet  Google Scholar 

  28. Codina R. Stabilized finite element approximation of transient incompressible flows using orthogonal subscales. Comput Methods Appl Mech Engrg, 2002, 191: 4295–4321

    Article  MathSciNet  Google Scholar 

  29. Crouzeix M, Raviart P-A. Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. RAIRO Sér Rouge, 1973, 7: 33–75

    MathSciNet  Google Scholar 

  30. Dauge M. Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. Part I. Linearized equations. SIAM J Math Anal, 1989, 20: 74–97

    Article  MathSciNet  Google Scholar 

  31. Di Pietro D A, Ern A. Discrete functional analysis tools for discontinuous Galerkin methods with application to the incompressible Navier-Stokes equations. Math Comp, 2010, 79: 1303–1330

    Article  MathSciNet  Google Scholar 

  32. Durán R G. Mixed finite element methods. In: Mixed Finite Elements, Compatibility Conditions, and Applications. Lecture Notes in Mathematics, vol. 1939. Berlin-Heidelberg: Springer, 2008, 1–44

    Google Scholar 

  33. Franca L P, Hughes T J R. Two classes of mixed finite element methods. Comput Methods Appl Mech Engrg, 1988, 69: 89–129

    Article  MathSciNet  Google Scholar 

  34. Fu G S, Jin Y Y, Qiu W F. Parameter-free superconvergent H(div)-conforming HDG methods for the Brinkman equations. IMA J Numer Anal, 2019, 39: 957–982

    Article  MathSciNet  Google Scholar 

  35. Gaël Guennebaud B J, et al. Eigen 3.2.5, https://eigen.tuxfamily.org/index.php?title=Main_Page, 2015

  36. Ganesan S, Matthies G, Tobiska L. Local projection stabilization of equal order interpolation applied to the Stokes problem. Math Comp, 2008, 77: 2039–2060

    Article  MathSciNet  Google Scholar 

  37. Gelhard T, Lube G, Olshanskii M A, et al. Stabilized finite element schemes with LBB-stable elements for incompressible flows. J Comput Appl Math, 2005, 177: 243–267

    Article  MathSciNet  Google Scholar 

  38. Girault V, Raviart P-A. Finite Element Approximation of the Navier-Stokes Equations. Lecture Notes in Mathematics, vol. 749. Berlin-New York: Springer-Verlag, 1979

    Book  Google Scholar 

  39. Girault V, Raviart P-A. Finite Element Methods for Navier-Stokes Equations. Springer Series in Computational Mathematics, vol. 5. Berlin: Springer-Verlag, 1986

    Book  Google Scholar 

  40. Gunzburger M D. Finite Element Methods for Viscous Incompressible Flows. Computer Science and Scientific Computing. Boston: Academic Press, 1989

    Google Scholar 

  41. John V, Linke A, Merdon C, et al. On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev, 2017, 59: 492–544

    Article  MathSciNet  Google Scholar 

  42. Karakashian O A, Jureidini W N. A nonconforming finite element method for the stationary Navier-Stokes equations. SIAM J Numer Anal, 1998, 35: 93–120

    Article  MathSciNet  Google Scholar 

  43. Kellogg R B, Osborn J E. A regularity result for the Stokes problem in a convex polygon. J Funct Anal, 1976, 21: 397–431

    Article  MathSciNet  Google Scholar 

  44. Lehrenfled C. Hybrid discontinuous Galerkin methods for solving incompressible flow problems. http://num.math.unigoettingen.de/~lehrenfeld/sections/pubs_src/Leh10_dina4.pdf, 2010

  45. Lehrenfeld C, Schöberl J. High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows. Comput Methods Appl Mech Engrg, 2016, 307: 339–361

    Article  MathSciNet  Google Scholar 

  46. Linke A. Divergence-free mixed finite elements for the incompressible Navier-Stokes equation. PhD Dissertation. Erlangen: University of Erlangen, 2007

    Google Scholar 

  47. Linke A. Collision in a cross-shaped domain—A steady 2d Navier-Stokes example demonstrating the importance of mass conservation in CFD. Comput Methods Appl Mech Engrg, 2009, 198: 3278–3286

    Article  MathSciNet  Google Scholar 

  48. Linke A, Merdon C. Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier-Stokes equations. Comput Methods Appl Mech Engrg, 2016, 311: 304–326

    Article  MathSciNet  Google Scholar 

  49. Lomtev I, Karniadakis G E. A discontinuous Galerkin method for the Navier-Stokes equations. Internat J Numer Methods Fluids, 1999, 29: 587–603

    Article  MathSciNet  Google Scholar 

  50. Matthies G, Skrzypacz P, Tobiska L. A unified convergence analysis for local projection stabilisations applied to the Oseen problem. ESAIM Math Model Numer Anal, 2007, 41: 713–742

    Article  MathSciNet  Google Scholar 

  51. Moro D, Nguyen N, Peraire J. Navier-Stokes solution using hybridizable discontinuous Galerkin methods. In: 20th AIAA Computational Fluid Dynamics Conference. Reston: AIAA, 2011, 3407

    Google Scholar 

  52. Nguyen N C, Peraire J, Cockburn B. A hybridizable discontinuous Galerkin method for Stokes flow. Comput Methods Appl Mech Engrg, 2010, 199: 582–597

    Article  MathSciNet  Google Scholar 

  53. Nguyen N C, Peraire J, Cockburn B. An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations. J Comput Phys, 2011, 230: 1147–1170

    Article  MathSciNet  Google Scholar 

  54. Olshanskii M A, Reusken A. Grad-div stabilization for Stokes equations. Math Comp, 2004, 73: 1699–1718

    Article  MathSciNet  Google Scholar 

  55. Peraire J, Nguyen N, Cockburn B. A hybridizable discontinuous Galerkin method for the compressible Euler and Navier-Stokes equations. In: 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition. Reston: AIAA, 2010, 363

    Google Scholar 

  56. Qiu W F, Shi K. A superconvergent HDG method for the incompressible Navier-Stokes equations on general polyhedral meshes. IMA J Numer Anal, 2016, 36: 1943–1967

    Article  MathSciNet  Google Scholar 

  57. Rhebergen S, Wells G N. A hybridizable discontinuous Galerkin method for the Navier-Stokes equations with pointwise divergence-free velocity field. J Sci Comput, 2018, 76: 1484–1501

    Article  MathSciNet  Google Scholar 

  58. Scott L R, Vogelius M. Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. ESAIM Math Model Numer Anal, 1985, 19: 111–143

    Article  MathSciNet  Google Scholar 

  59. Taylor C, Hood P. A numerical solution of the Navier-Stokes equations using the finite element technique. Comput & Fluids, 1973, 1: 73–100

    Article  MathSciNet  Google Scholar 

  60. Waluga C. Analysis of hybrid discontinuous Galerkin methods for incompressible flow problems. PhD Thesis. Aachen: RWTH Aachen University, 2012

    Google Scholar 

  61. Xie X-P, Xu J C, Xue G G. Uniformly-stable finite element methods for Darcy-Stokes-Brinkman models. J Comput Math, 2008, 26: 437–455

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The first author was supported by National Natural Science Foundation of China (Grant Nos. 12171341 and 11801063) and the Fundamental Research Funds for the Central Universities (Grant No. YJ202030). The second author was supported by National Natural Science Foundation of China (Grant Nos. 12171340 and 11771312).

Author information

Authors and Affiliations

  1. School of Mathematics, Sichuan University, Chengdu, 610064, China

    Gang Chen & Xiaoping Xie

Authors
  1. Gang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaoping Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaoping Xie.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, G., Xie, X. Analysis of a class of globally divergence-free HDG methods for stationary Navier-Stokes equations. Sci. China Math. 67, 1133–1158 (2024). https://doi.org/10.1007/s11425-022-2077-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-022-2077-7

Keywords

MSC(2020)

Search

Navigation