Skip to main content
SpringerLink
Log in

A three-step defect-correction stabilized algorithm for incompressible flows with non-homogeneous Dirichlet boundary conditions

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

Based on two-grid discretizations and quadratic equal-order finite elements for the velocity and pressure approximations, we develop a three-step defect-correction stabilized algorithm for the incompressible Navier-Stokes equations, where non-homogeneous Dirichlet boundary conditions are considered and high Reynolds numbers are allowed. In this developed algorithm, we first solve an artificial viscosity stabilized nonlinear problem on a coarse grid in a defect step and then correct the resulting residual by solving two stabilized and linearized problems on a fine grid in correction steps. While the fine grid correction problems have the same stiffness matrices with only different right-hand sides. We use a variational multiscale method to stabilize the system, making the algorithm has a broad range of potential applications in the simulation of high Reynolds number flows. Under the weak uniqueness condition, we give a stability analysis of the present algorithm, analyze the error bounds of the approximate solutions, and derive the algorithmic parameter scalings. Finally, we perform a series of numerical examples to demonstrate the promise of the proposed algorithm.

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

Similar content being viewed by others

Availability of data

Data sharing is not applicable to this article as no databases were generated or analyzed during the study.

References

  1. Wang, K.: Iterative schemes for the non-homogeneous Navier-Stokes equations based on the finite element approximation. Comput. Math. Appl. 71, 120–132 (2016)

    Article  MathSciNet  Google Scholar 

  2. Girault, V., Raviart, P.A.: Finite element approximation of the Navier-Stokes equations. Springer-Verlag, Berlin, Heidelberg (1979)

    Book  Google Scholar 

  3. Temam, R.: Navier-Stokes equations: theory and numerical analysis. North-Holland, Amsterdam (1984)

    Google Scholar 

  4. Girault, V., Raviart, P.A.: Finite element method for Navier-Stokes equations: theory and algorithms. Springer-Verlag, Berlin, Heidelberg (1986)

    Book  Google Scholar 

  5. John, V.: Finite element methods for incompressible flow problems. Springer International Publishing (2016)

  6. 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 

  7. Ervin, V.J., Layton, W.J., Maubach, J.M.: Adaptive defect-correction methods for viscous incompressible flow problems. SIAM J. Numer. Anal. 37, 1165–1185 (2000)

    Article  MathSciNet  Google Scholar 

  8. Layton, W.J., Lee, H., Peterson, J.: A defect-correction method for the incompressible Navier-Stokes equations. Appl. Math. Comput. 129, 1–19 (2002)

    MathSciNet  Google Scholar 

  9. Labovschii, A.: A defect correction method for the time-dependent Navier-Stokes equations. Numer. Methods Partial Differ. Equ. 25, 1–25 (2009)

    Article  MathSciNet  Google Scholar 

  10. Wang, K.: A new defect correction method for the Navier-Stokes equations at high Reynolds numbers. Appl. Math. Comput. 216, 3252–3264 (2010)

    MathSciNet  ADS  Google Scholar 

  11. Si, Z.Y.: Second order modified method of characteristics mixed defect-correction finite element method for time-dependent Navier-Stokes problems. Numer. Algor. 59, 271–300 (2012)

    Article  MathSciNet  Google Scholar 

  12. 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 

  13. Huang, P.Z., Feng, X.L., He, Y.N.: Two-level defect-correction Oseen iterative stabilized finite element methods for the stationary Navier-Stokes equations. Appl. Math. Model. 37, 728–741 (2013)

    Article  MathSciNet  Google Scholar 

  14. Qiu, H.L., Mei, L.Q.: Two-level defect-correction stabilized finite element method for Navier-Stokes equations with friction boundary conditions. J. Comput. Appl. Math. 280, 80–93 (2015)

    Article  MathSciNet  Google Scholar 

  15. Wen, J., He, Y.N.: A new defect-correction method for the stationary Navier-Stokes equations based on pressure projection. Math. Methods Appl. Sci. 41, 250–260 (2018)

    Article  MathSciNet  ADS  Google Scholar 

  16. Shang, Y.Q.: A new two-level defect-correction method for the steady Navier-Stokes equations. J. Comput. Appl. Math. 381, 113009 (2020)

    Article  MathSciNet  Google Scholar 

  17. Guermond, J.: Stabilization of Galerkin approximation of transport equations by subgrid modeling. M2AN Math. Moder. Numer. Anal. 33, 1293-1316 (1999)

  18. Layton, W.J.: A connection between subgrid scale eddy viscosity and mixed methods. Appl. Math. Comput. 133, 147–157 (2002)

    MathSciNet  Google Scholar 

  19. Kaya, S., Layton, W.J., Riviere, B.: Subgrid stabilized defect correction methods for the Navier-Stokes equations. SIAM J. Numer. Anal. 44, 1639–1654 (2006)

    Article  MathSciNet  Google Scholar 

  20. Guermond, J., Marra, A., Quartapelle, L.: Subgrid stabilized projection method for 2D unsteady flows at high Reynolds numbers. Comput. Methods Appl. Mech. Engrg. 195, 5857–5876 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  21. Zhang, Y., He, Y.N.: Assessment of subgrid-scale models for the incompressible Navier-Stokes equations. J. Comput. Appl. Math. 234, 593–604 (2010)

    Article  MathSciNet  Google Scholar 

  22. Galvin, K.J.: New subgrid artificial viscosity Galerkin methods for the Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 200, 242–250 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  23. Shang, Y.Q.: A two-level subgrid stabilized Oseen iterative method for the steady Navier-Stokes equations. J. Comput. Phys. 233, 210–226 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  24. Shang, Y.Q., Huang, S.M.: A parallel subgrid stabilized finite element method based on two-grid discretization for simulation of 2D/3D steady incompressible flows. J. Sci. Comput. 60, 564–583 (2014)

    Article  MathSciNet  Google Scholar 

  25. Shang, Y.Q., Qin, J.: A two-parameter stabilized finite element method for incompressible flows. Numer. Methods Partial Differ. Equ. 3, 425–444 (2017)

    Article  MathSciNet  Google Scholar 

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

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

    Article  MathSciNet  ADS  Google Scholar 

  28. Brooks, 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. Engrg. 32, 199–259 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  29. Hughes, T.J.R.: Multiscale phenomena: green’s functions, the Dirichlet-to-Neumann formulation, subgrid-scale models, bubbles and the origins of stabilized methods. Comput. Methods Appl. Mech. Engrg. 127, 387–401 (1995)

    Article  MathSciNet  ADS  Google Scholar 

  30. Hughes, T.J.R., Mazzei, L., Oberai, A.A.: The multiscale formulation of large eddy simulation: decay of homogeneous isotropic turbulence. Phys. Fluids 13, 505–511 (2001)

    Article  CAS  ADS  Google Scholar 

  31. John, V., Kaya, S.: A finite element variational multiscale method for the Navier-Stokes equations. SIAM J. Sci. Comput. 26, 1485–1503 (2005)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  33. 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  ADS  Google Scholar 

  34. Zheng, H.B., Hou, Y.R., Shi, F., Song, L.N.: A finite element variational multiscale method for incompressible flows based on two local Gauss integrations. J. Comput. Phys. 228, 5961–5977 (2009)

    Article  MathSciNet  CAS  ADS  Google Scholar 

  35. Zheng, H.B., Hou, Y.R., Shi, F.: Adaptive finite element variational multiscale method for incompressible flows based on two local Gauss integrations. J. Comput. Phys. 229, 7030–7041 (2010)

    Article  MathSciNet  CAS  ADS  Google Scholar 

  36. Shi, F., Zheng, H.B., Yu, J.P., Li, Y.: On the convergence of variational multiscale methods based on Newton’s iteration for the incompressible flows. Appl. Math. Model. 38, 5726–5742 (2014)

    Article  MathSciNet  Google Scholar 

  37. Shang, Y.Q.: A parallel two-level variational multiscale method for the Navier-Stokes equations. Nonlineal Anal. 84, 103–116 (2013)

    Article  MathSciNet  Google Scholar 

  38. Shang, Y.Q., Qin, J.: A finite element variational multiscale method based on two-grid discretization for the steady incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 300, 182–198 (2016)

    Article  MathSciNet  ADS  Google Scholar 

  39. Zheng, B., Shang, Y.Q.: A parallel stabilized finite element variational multiscale method based on fully overlapping domain decomposition for the incompressible Navier-Stokes equations. Appl. Numer. Math. 159, 138–158 (2021)

    Article  MathSciNet  Google Scholar 

  40. Xu, J.C.: A novel two-grid method for semilinaer elliptic equations. SIAM J. Sci. Comput. 15, 231–237 (1994)

    Article  MathSciNet  Google Scholar 

  41. Xu, J.C.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33, 1759–1777 (1998)

    Article  MathSciNet  Google Scholar 

  42. Layton, W.J.: A two level discretization method for the Navier-Stokes equations. Comput. Math. Appl. 5, 33–38 (1993)

    Article  MathSciNet  Google Scholar 

  43. Layton, W.J., Lenferink, H.W.J.: A two-level method with backtracking for the Navier-Stokes equations. SIAM J. Numer. Anal. 35, 2035–2054 (1998)

    Article  MathSciNet  Google Scholar 

  44. Layton, W.J., Lee, H.E., Peterson, J.: Numerical solution of the stationary Navier-Stokes equations using a multilevel finite element method. SIAM J. Sci. Comput. 20, 1–12 (1998)

    Article  MathSciNet  Google Scholar 

  45. Layton, W.J., Lenferink, H.W.J.: A multilevel mesh independence principle for the Navier-Stokes equations. SIAM J. Numer. Anal. 33, 17–30 (1996)

    Article  MathSciNet  Google Scholar 

  46. Franca, L.P., Nesliturk, A.: On a two-level finite element method for the incompressible Navier-Stokes equations. Int. J. Numer. Methods Engrg. 52, 433–453 (2001)

    Article  Google Scholar 

  47. He, Y.N.: Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations. SIAM J. Numer. Anal. 41, 1263–1285 (2003)

    Article  MathSciNet  Google Scholar 

  48. He, Y.N., Liu, K.M., Sun, W.W.: Multi-level spectral Galerkin method for the Navier-Stokes equations I: spatial discretization. Numer. Math. 101, 501–522 (2005)

    Article  MathSciNet  Google Scholar 

  49. He, Y.N., Liu, K.M.: Multi-level spectral Galerkin method for the Navier-Stokes equations, II: time discretization. Adv. Comput. Math. 25, 403–433 (2006)

    Article  MathSciNet  Google Scholar 

  50. He, Y.N., Wang, Y.W: A simplified two-level method for the steady Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 197, 1568-1576 (2008)

  51. Dai, X.X., Cheng, X.L.: A two-grid method based on Newton iteration for the Navier-Stokes equations. J. Comput. Appl. Math. 220, 566–573 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  52. He, Y.N., Mei, L.Q., Shang, Y.Q., Cui, J.: Newton iterative parallel finite element algorithm for the steady Navier-Stokes equations. J. Sci. Comput. 44, 92–106 (2010)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  55. Zheng, H.B., Yu, J.P., Shi, F.: Local and parallel finite element method based on the partition of unity for incompressible flow. J. Sci. Comput. 65, 512–532 (2015)

    Article  MathSciNet  Google Scholar 

  56. 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. 75, 1445–1462 (2018)

    Article  MathSciNet  Google Scholar 

  57. Li, J.: Investigations on two kinds of two-level stabilized finite element methods for the stationary Navier-Stokes equations. Appl. Math. Comput. 182, 140–1481 (2006)

    MathSciNet  Google Scholar 

  58. Li, J., He, Y.N., Xu, H.: A multi-level stabilized finite element method for the stationary Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 196, 2852–2862 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  59. 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 

  60. Qiu, H.Q.: Two-grid stabilized methods for the stationary incompressible Navier-Stokes equations with nonlinear slip boundary conditions. Appl. Math. Comput. 332, 172–188 (2018)

    MathSciNet  Google Scholar 

  61. Borggaard, J., Iliescu, T., Lee, H., Roop, J.P. et al.: A two-level discretization method for the Smagorinsky model. Multiscale Model. Simul. 7, 599–621 (2008)

  62. Huang, P.Z., Feng, X.L.: Two-level stabilized method based on Newton iteration for the steady Smagorinsky model. Nonlinear Anal. Real World Appl. 14, 1795–1805 (2013)

    Article  MathSciNet  Google Scholar 

  63. Qiu, H.L., Mei, L.Q.: Multi-level stabilized algorithms for the stationary incompressible Navier-Stokes equations with damping. Appl. Numer. Math. 143, 188–202 (2019)

    Article  MathSciNet  Google Scholar 

  64. Li, M.H., Shi, D.Y., Li, Z.Z., Chen, H.R.: Two-level mixed finite element methods for the Navier-Stokes equations with damping. J. Math. Anal. Appl. 470, 292–307 (2019)

    Article  MathSciNet  Google Scholar 

  65. 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, 1180–1190 (2010)

    Article  MathSciNet  Google Scholar 

  66. 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 

  67. Li, Z.Z., Shi, D.Y., Li, M.H.: Stabilized mixed finite element methods for the Navier-Stokes equations with damping. Math. Methods Appl. Sci. 42, 605–619 (2019)

    Article  MathSciNet  ADS  Google Scholar 

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

    Google Scholar 

  69. 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. Engrg. 198, 1351–1359 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  70. Maubach, J.: Local bisection refinement for n-simplicial grids generated by reflection. SIAM J. Sci. Comput. 16, 210–227 (1995)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  72. Erturk, E., Corke, T.C., Gokcol, C.: Numerical solutions of 2-d steady incompressible driven cavity flow at high Reynolds numbers. Int. J. Numer. Methods Fluids 48, 747–774 (2005)

    Article  ADS  Google Scholar 

  73. Gartling, D.K.: A test problem for outflow boundary conditions-flow over a backward-facing step. Int. J. Numer. Methods Fluids 11, 953–967 (1990)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the editors and anonymous reviewers for their valuable comments and suggestions, which led to an improvement in the paper.

Funding

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

Author information

Authors and Affiliations

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

    Bo Zheng & Yueqiang Shang

Authors
  1. Bo Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yueqiang Shang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yueqiang Shang.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Communicated by: Jon Wilkening

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zheng, B., Shang, Y. A three-step defect-correction stabilized algorithm for incompressible flows with non-homogeneous Dirichlet boundary conditions. Adv Comput Math 50, 3 (2024). https://doi.org/10.1007/s10444-023-10101-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10444-023-10101-8

Keywords

Mathematics Subject Classification (2010)

Search

Navigation