Skip to main content
Springer Nature Link
Log in

A Parallel Partition of Unity Scheme Based on Two-Grid Discretizations for the Navier–Stokes Problem

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

A parallel partition of unity scheme based on two-grid discretizations is proposed and analyzed in this paper for the Navier–Stokes problem. A standard Galerkin finite element method on a relatively coarse grid is used to obtain the approximation of the lower frequency components and the higher frequency components are computed on fine grids by some local and parallel procedure. The motivation of the proposed parallel partition of unity scheme is based on the superposition principle. That is the original linear problem, which is a linear residual equation, can be equivalent to the sum of a series of simple problems of same type with free terms of small supports. Each simple problem is approximated by a local problem with homogeneous Dirichlet boundary condition. Optimal error estimates are obtained and some numerical tests are presented to support the theoretical results.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. Babus̆ka, I., Melenk, J.: The partition of unity method. Int. J. Numer. Meth. Eng. 40(4), 727–758 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bacuta, C., Sun, J., Zheng, C.: Partition of unity refinement for local approximation. Numer. Methods Partial Differ. Equ. 27(4), 803–817 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bank, R., Holst, M.: A new paradigm for parallel adaptive meshing algorithms. SIAM J. Sci. Comput. 22(4), 1411–1443 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bank, R., Holst, M.: A new paradigm for parallel adaptive meshing algorithms. SIAM Rev. 45(2), 291–323 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ciarlet, P.G., Lions,J.L.: Handbook of Numerical Analysis, Vol. II, Finite Element Methods (Part I), North-Holland, Amsterdam (1991)

  7. Du, G., Hou, Y.: A parallel partition of unity method for the time-dependent convection-diffusion equations. Int. J. Numer. Meth. Heat Fluid Flow 25(8), 1947–1956 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Du, G., Hou, Y., Zuo, L.: Local and parallel finite element methods for the mixed Navier-Stokes/Darcy model. Int. J. Comput. Math. 93(7), 1155–1172 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Du, G., Hou, Y., Zuo, L.: A modified local and parallel finite element method for the mixed Stokes-Darcy model. J. Math. Anal. Appl. 435(2), 1129–1145 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Erturk, E., Corke, E.T., Gokcol, C.: Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers. Int. J. Numer. Methods Fluids 48(7), 747–774 (2005)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Ghia, U., Ghia, K.N., Shin, C.T.: High-Re solutions for incompressible flow using the Nacier–Stokes equations and a multigrid method. J. Comput. Phys. 48(3), 387–411 (1982)

    Article  MATH  Google Scholar 

  13. Girault, V., Glowinski, R., López, H., Vila, J.-P.: A boundary multiplier/fictitious domain method for the steady incompressible Navier–Stokes equations. Numer. Math. 88(75), 75–103 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. He, Y., Mei, L., Shang, Y., Cui, J.: Newton iterative parallel finite element algorithm for the steady Navier–Stokes equations. J. Sci. Comput. 44(1), 92–106 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. He, Y., Xu, J., Zhou, A.: Local and parallel finite element algorithms for the Stokes problem. Numer. Math. 109(3), 415–434 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Holst, M., Holst, M.: Adaptive numerical treatment of elliptic systems on manifolds. Adv. Comput. Math. 15(1), 139–191 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Holst, M.: Applications of domain decomposition and partition of unity methods in physics and geometry (plenary paper). In Herrera, I., Keyes, D.E., Widlund, O.B., Yates, R. (eds), Proceedings of the Fourteenth International Conference on Domain Decomposition Methods, The National Autonomous University of Mexico (UNAM), Mexico City, pp. 63–78 (2003)

  19. Hou, Y., Du, G.: An expandable local and parallel two-grid finite element scheme. Comput. Math. Appl. 71(12), 2541–2556 (2016)

    Article  MathSciNet  Google Scholar 

  20. Huang, Y., Xu, J.: A conforming finite element method for overlapping and nonmatching grids. Math. Comput. 72(243), 1057–1066 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Larson, M.G., Malqvist, A.: Adaptive variational multi-scale methods based on a posteriori error estimation: energy norm estimates for elliptic problems. Comput. Methods Appl. Mech. Eng. 196(21–24), 2313–2324 (2007)

    Article  MATH  Google Scholar 

  22. Larson, M.G., Malqvist, A.: An adaptive variational multi-scale method for convection-diffusion problems. Commun. Numer. Methods Eng. 25(1), 65–79 (2009)

    Article  Google Scholar 

  23. Li, K., Hou, Y.: An AIM and one-step newton method for the Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 190(46), 6141–6155 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Liu, Q., Hou, Y.: Local and parallel finite element algorithms for time-dependent convection-diffusion equations. Appl. Math. Mech. 30(6), 787–794 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Melenk, J., Babus̆ka, I.: The partition of unity finite element method: basic theory and applications. Comput. Methods Appl. Mech. Eng. 139(1–4), 289–314 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. Song, L., Hou, Y., Zheng, H.: Adaptive local postprocessing finite element method for the Navier–Stokes equations. J. Sci. Comput. 55(2), 255–267 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Xie, C., Zheng, H.: A parallel variational multiscale method for incompressible flows based on the partition of unity. Int. J. Numer. Anal. Model. 11(4), 854–865 (2014)

    MathSciNet  Google Scholar 

  28. Xu, J., Zhou, A.: Some local and parallel properties of finite element discretizations. In Proceedings for Eleventh International Conference on Domain Decomposition Methods, pp. 140–147 (1999)

  29. Xu, J., Zhou, A.: Local and parallel finite element algorithms for eigenvalue problems. Acta Math. Appl. Sin. 18, 185–200 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  30. Xu, J., Zhou, A.: Local and parallel finite element algorithms based on two-grid discretizations. Math. Comput. 69(231), 881–909 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  31. Xu, J., Zhou, A.: Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problems. Adv. Comput. Math. 14(4), 293–327 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yu, J., Shi, F., Zheng, H.: Local and parallel finite element method based on the partition of unity for the stokes problem. SIAM J. Sci. Comput. 36(5), C547–C567 (2014)

    Article  MATH  Google Scholar 

  33. Zheng, H., Song, L., Hou, Y., Zhang, Y.: The partition of unity parallel finite element algorithm. Adv. Comput. Math. 41(4), 937–951 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zheng, H., Yu, J., Shi, F.: Local and parallel finite element method based on the partition of unity for incompressible flow. J. Sci. Comput. 65(2), 512–532 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, Shandong, 250014, China

    Guangzhi Du

  2. School of Mathematical Sciences, University of Jinan, Jinan, Shandong, 250022, China

    Liyun Zuo

Authors
  1. Guangzhi Du
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Liyun Zuo
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Guangzhi Du.

Additional information

G. Du: Subsidized by NSFC (Grant No. 11701343) and partially supported by NSFC (Grant Nos. 11571274, 11401466)

L. Zuo: Subsidized by the Provincial Natural Science Foundation of Shandong (Grant No. ZR2017BA027).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Du, G., Zuo, L. A Parallel Partition of Unity Scheme Based on Two-Grid Discretizations for the Navier–Stokes Problem. J Sci Comput 75, 1445–1462 (2018). https://doi.org/10.1007/s10915-017-0593-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-017-0593-6

Keywords

Mathematics Subject Classification