Skip to main content
SpringerLink
Log in

Explicit a posteriori and a priori error estimation for the finite element solution of Stokes equations

  • Original Paper
  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

For the Stokes equation over 2D and 3D domains, explicit a posteriori and a priori error estimation are novelly developed for the finite element solution. The difficulty in handling the divergence-free condition of the Stokes equation is solved by utilizing the extended hypercircle method along with the Scott-Vogelius finite element scheme. Since all terms in the error estimation have explicit values, by further applying the interval arithmetic and verified computing algorithms, the computed results provide rigorous estimation for the approximation error. As an application of the proposed error estimation, the eigenvalue problem of the Stokes operator is considered and rigorous bounds for the eigenvalues are obtained. The efficiency of proposed error estimation is demonstrated by solving the Stokes equation on both convex and non-convex 3D domains.

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.

Similar content being viewed by others

References

  1. Balay,S., Gropp,W.D., McInnes, L.C., Smith,B.F.: Efficient management of parallelism in object oriented numerical software libraries. In E. Arge, A. M. Bruaset, and H. P. Langtangen, editors, Modern Software Tools in Scientific Computing, pages 163–202. Birkhäuser Press, (1997)

  2. Boffi, Daniele, Brezzi, Franco, Fortin, Michel, et al.: Mixed finite element methods and applications, vol. 44. Springer (2013)

    Book  Google Scholar 

  3. Cott,L R S., Ogelius, M V., Scott,L R.,  Vogelius,M.: Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. ESAIM: Mathematical Modelling and Numerical Analysis, 19(1):111–143, (1985)

  4. Girault,V., Raviart,P-A.: Finite element methods for Navier-Stokes equations: theory and algorithms, volume 5. Springer Science & Business Media, (2012)

  5. Hernandez, V., Roman, J.E., Vidal, V.: SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Software 31(3), 351–362 (2005)

    Article  MathSciNet  Google Scholar 

  6. Kikuchi, F., Saito, H.: Remarks on a posteriori error estimation for finite element solutions. J. Comput. Appl. Math. 199, 329–336 (2007)

    Article  MathSciNet  Google Scholar 

  7. Li, Q., Liu, X.: Explicit finite element error estimates for nonhomogeneous neumann problems. Appl. of Math. 63(3), 367–379 (2018)

    Article  MathSciNet  Google Scholar 

  8. Liu, X.: A framework of verified eigenvalue bounds for self-adjoint differential operators. Appl. Math. & Comput. 267, 341–355 (2015)

    MathSciNet  MATH  Google Scholar 

  9. Liu, X.: Explicit eigenvalue bounds of differential operators defined by symmetric positive semi-definite bilinear forms. J. Comput. & App. Math. 371, 112666 (2020)

    Article  MathSciNet  Google Scholar 

  10. Liu, X.: Stokes eigenvalue problem. Online scientific computing platform, created on July, Ganjin (2020). https://ganjin.online/xfliu/StokesEigProblem

  11. Liu, X., Oishi, S.: Verified eigenvalue evaluation for the Laplacian over polygonal domains of arbitrary shape. SIAM J. Numer. Anal. 51(3), 1634–1654 (2013)

    Article  MathSciNet  Google Scholar 

  12. Nakao, M.T., Yamamoto, N., Watanabe, Y.: A posteriori and constructive a priori error bounds for finite element solutions of the Stokes equations. J. Comput. & Appl. Math. 91(1), 137–158 (1998)

    Article  MathSciNet  Google Scholar 

  13. Rump,S.M.: Intlab - interval laboratory. In Tibor Csendes, editor, Developments in Reliable Computing, pages 77–104. Kluwer Academic Publishers, Dordrecht, (1999)

  14. Stenberg, Rolf: Some new families of finite elements for the stokes equations. Numerische Mathematik 56(8), 827–838 (1989)

    Article  MathSciNet  Google Scholar 

  15. Verfürth, R.: A posteriori error estimators for the stokes equations. Numer. Math. 55(3), 309–325 (1989)

    Article  MathSciNet  Google Scholar 

  16. Watanabe, Y., Yamamoto, N., Nakao, M.T.: A numerical verification method of solutions for the Navier-Stokes equations. Reliab. Comput. 5(3), 347–357 (1999)

    Article  MathSciNet  Google Scholar 

  17. Xie, M., Xie, H., Liu, X.: Explicit lower bounds for Stokes eigenvalue problems by using nonconforming finite elements. Jpn. J. Ind. Appl. Math. 35(1), 335–354 (2018)

    Article  MathSciNet  Google Scholar 

  18. Zhang, S.: On the divergence-free finite element method for the Stokes equations and the P1 Powell-Sabin divergence-free element. Math. of Comput. 74(250), 543–554 (2004)

    Article  Google Scholar 

  19. Zhang, S.: A new family of stable mixed finite elements for the 3D Stokes equations. Math. of Comput. 74(250), 543–554 (2005)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Science and Technology, Niigata University, Niigata, 950-2181, Japan

    Xuefeng Liu

  2. Faculty of Science and Engineering, Waseda University, Tokyo, 169-8555, Japan

    Mitsuhiro T. Nakao & Shin’ichi Oishi

  3. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

    Chun’guang You

Authors
  1. Xuefeng Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mitsuhiro T. Nakao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chun’guang You
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Shin’ichi Oishi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xuefeng Liu.

Additional information

Publisher's Note

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

The first author is supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B) 16H03950, 20H01820 and Grant-in-Aid for Scientific Research (C) 18K03411. The second author is supported by Grant-in-Aid for Scientific Research (C) 18K03434. The last author is supported by JST CREST Grant Number JPMJCR14D4, Japan.

The third author is Doctor course student in this affliction when he joined this research.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, X., Nakao, M.T., You, C. et al. Explicit a posteriori and a priori error estimation for the finite element solution of Stokes equations. Japan J. Indust. Appl. Math. 38, 545–559 (2021). https://doi.org/10.1007/s13160-020-00449-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13160-020-00449-5

Keywords

Mathematics Subject Classification

Search

Navigation