Skip to main content
SpringerLink
Log in

A Modified Rotated-\(Q_1\) Finite Element for the Stokes Equations on Quadrilateral and Hexahedral Meshes

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

Abstract

We construct a modified rotated-\(Q_1\) finite element, where we replace the \(P_2\) bubbles of the Rannacher–Turek rotated-\(Q_1\) element by multi-piece linear polynomials. In 2D, we use \(\{1,x,y,|\lambda _1|\}\) as the basis on a general quadrilateral, where \(\lambda _1\) is a linear polynomial which vanishes at the middle edge \(\textbf{x}_3\textbf{x}_4\) of two opposite mid-edge nodes \(\textbf{x}_1\) and \(\textbf{x}_2\). In 3D, we use \(\{1,x,y,z, |\lambda _1|,|\lambda _3|\}\) as the basis on a general hexahedron, where \(\lambda _1\) is a linear polynomial which vanishes at the mid plane \(\textbf{x}_3\textbf{x}_4\textbf{x}_5\textbf{x}_6\) between the two opposite mid-face nodes \(\textbf{x}_1\) and \(\textbf{x}_2\), and \(\lambda _3\) is a linear polynomial which vanishes at the mid plane \(\textbf{x}_1\textbf{x}_2\textbf{x}_5\textbf{x}_6\) between the two opposite mid-face nodes \(\textbf{x}_3\) and \(\textbf{x}_4\). The new rotated-\(Q_1\) finite element is shown inf-sup stable and quasi-optimal in solving the Stokes equations, on general quadrilateral and hexahedral meshes. Numerical tests in 2D and 3D show the method does converge quasi-optimally on non-asymptotic parallelogram or parallelepiped meshes, while the Rannacher–Turek rotated-\(Q_1\) element fails to converge on such meshes.

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
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Data Availability

This research does not use any external or author-collected data.

References

  1. Ainsworth, M.: A posteriori error estimation for non-conforming quadrilateral finite elements. Int. J. Numer. Anal. Model. 2(1), 1–18 (2005)

    MathSciNet  Google Scholar 

  2. Arnold, D.N., Qin, J.: Quadratic velocity/linear pressure Stokes elements. In: Vichnevetsky, R., Steplemen, R.S. (eds) Advances in Computer Methods for Partial Differential Equations VII (1992)

  3. Chen, J.R., Xu, X.J.: The mortar element method for rotated Q1 element. J. Comput. Math. 20(3), 313–324 (2002)

    MathSciNet  Google Scholar 

  4. Chen,J., Xu,X.: The mortar element method for the rotated Q1 element, Domain decomposition methods in science and engineering (Lyon, 2000), 223–228, Theory Eng. Appl. Comput. Methods, Internat. Center Numer. Methods Eng. (CIMNE), Barcelona (2002)

  5. Crouzeix, M., Raviart, P.A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Modél. Math. Anal. Numér. 3, 33–75 (1973)

    MathSciNet  Google Scholar 

  6. Fabien, M., Guzman, J., Neilan, M., Zytoon, A.: Low-order divergence-free approximations for the Stokes problem on Worsey–Farin and Powell–Sabin splits. Comput. Methods Appl. Mech. Eng. 390, 114444 (2022)

    Article  MathSciNet  Google Scholar 

  7. Falk, R., Neilan, M.: Stokes complexes and the construction of stable finite elements with pointwise mass conservation. SIAM J. Numer. Anal. 51(2), 1308–1326 (2013)

    Article  MathSciNet  Google Scholar 

  8. Guzman, J., Neilan, M.: Conforming and divergence-free Stokes elements on general triangular meshes. Math. Comput. 83(285), 15–36 (2014)

    Article  MathSciNet  Google Scholar 

  9. Guzman, J., Neilan, M.: Conforming and divergence-free Stokes elements in three dimensions. IMA J. Numer. Anal. 34(4), 1489–1508 (2014)

    Article  MathSciNet  Google Scholar 

  10. Guzman, J., Neilan, M.: Inf-sup stable finite elements on barycentric refinements producing divergence-free approximations in arbitrary dimensions. SIAM J. Numer. Anal. 56(5), 2826–2844 (2018)

    Article  MathSciNet  Google Scholar 

  11. Huang, J., Zhang, S.: A divergence-free finite element method for a type of 3D Maxwell equations. Appl. Numer. Math. 62(6), 802–813 (2012)

    Article  MathSciNet  Google Scholar 

  12. Huang, P., Wang, F., Chen, J.: Cascadic multigrid method for mortar-type rotated Q1 element. J. Nanjing Norm. Univ. Nat. Sci. Ed. 30(4), 20–27 (2007)

    MathSciNet  Google Scholar 

  13. Huang, Y., Zhang, S.: A lowest order divergence-free finite element on rectangular grids. Front. Math. China 6(2), 253–270 (2011)

    Article  MathSciNet  Google Scholar 

  14. Huang, Y., Zhang, S.: Supercloseness of the divergence-free finite element solutions on rectangular grids. Commun. Math. Stat. 1(2), 143–162 (2013)

    Article  MathSciNet  Google Scholar 

  15. Kean, K., Neilan, M., Schneier, M.: The Scott–Vogelius method for the Stokes problem on anisotropic meshes. Int. J. Numer. Anal. Model. 19(2–3), 157–174 (2022)

    MathSciNet  Google Scholar 

  16. Neilan, M.: Discrete and conforming smooth de Rham complexes in three dimensions. Math. Comput. 84(295), 2059–2081 (2015)

    Article  MathSciNet  Google Scholar 

  17. Neilan, M., Otus, B.: Divergence-free Scott–Vogelius elements on curved domains. SIAM J. Numer. Anal. 59(2), 1090–1116 (2021)

    Article  MathSciNet  Google Scholar 

  18. Neilan, M., Sap, D.: Stokes elements on cubic meshes yielding divergence-free approximations. Calcolo 53(3), 263–283 (2016)

    Article  MathSciNet  Google Scholar 

  19. Qin, J.: On the convergence of some low order mixed finite elements for incompressible fluids. Pennsylvania State University, Thesis (1994)

  20. Qin, J., Zhang, S.: Stability and approximability of the P1–P0 element for Stokes equations. Int. J. Numer. Methods Fluids 54, 497–515 (2007)

    Article  Google Scholar 

  21. Rannacher, R., Turek, S.: Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differ. Equ. 8(2), 97–111 (1992)

    Article  MathSciNet  Google Scholar 

  22. Scott, L.R., Vogelius, M.: Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials, RAIRO. Modelisation Math. Anal. Numer. 19, 111–143 (1985)

    Article  MathSciNet  Google Scholar 

  23. Scott, L.R., Vogelius, M.: Conforming finite element methods for incompressible and nearly incompressible continua. Lectures in Applied Mathematics 22, pp. 221–244 (1985)

  24. Scott, L.R., Zhang, S.: Higher-dimensional nonnested multigrid methods. Math. Comput. 58(198), 457–466 (1992)

    Article  MathSciNet  Google Scholar 

  25. Shi, D., Liu, Q.: Nonconforming quadrilateral EQrot1 finite element method for Ginzburg–Landau equation. Numer. Methods Partial Differ. Equ. 36(2), 329–341 (2020)

    Article  Google Scholar 

  26. Xu, X., Zhang, S.: A new divergence-free interpolation operator with applications to the Darcy–Stokes–Brinkman equations. SIAM J. Sci. Comput. 32(2), 855–874 (2010)

    Article  MathSciNet  Google Scholar 

  27. Ye, X., Zhang, S.: A numerical scheme with divergence free H-div triangular finite element for the Stokes equations. Appl. Numer. Math. 167, 211–217 (2021)

    Article  MathSciNet  Google Scholar 

  28. Zhang, M., Luo, K., Zhang, S.Q.: Nonconforming-conforming finite element method for 3D Stokes problem. Sichuan Daxue Xuebao 55(1), 37–41 (2018)

    MathSciNet  Google Scholar 

  29. Zhang, M., Zhang, S.: A 3D conforming-nonconforming mixed finite element for solving symmetric stress Stokes equations. Int. J. Numer. Anal. Model. 14(4–5), 730–743 (2017)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  31. Zhang, S.: On the P1 Powell–Sabin divergence-free finite element for the Stokes equations. J. Comput. Math. 26, 456–470 (2008)

    MathSciNet  Google Scholar 

  32. Zhang, S.: A family of \(Q_ k+1, k \times Q_k, k+1 \) divergence-free finite elements on rectangular grids. SIAM J. Num. Anal. 47, 2090–2107 (2009)

    Article  Google Scholar 

  33. Zhang, S.: Quadratic divergence-free finite elements on Powell–Sabin tetrahedral grids. Calcolo 48(3), 211–244 (2011)

    Article  MathSciNet  Google Scholar 

  34. Zhang, S.: Divergence-free finite elements on tetrahedral grids for \(k\ge 6\). Math. Comput. 80, 669–695 (2011)

    Article  Google Scholar 

  35. Zhang, S.: A P4 bubble enriched P3 divergence-free finite element on triangular grids. Comput. Math. Appl. 74(11), 2710–2722 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

None.

Funding

The work of Liwei Xu is supported in part by NSFC Grant 12071060. Xuejun Xu was supported by National Natural Science Foundation of China (Grant No. 12071350), Shanghai Municipal Science and Technology Major Project No. 2021SHZDZX0100, and Science and Technology Commission of Shanghai Municipality.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, Sichuan, China

    Liwei Xu

  2. School of Mathematical Science, Tongji University, Shanghai, 200092, China

    Xuejun Xu

  3. Institute of Computational Mathematics, AMSS, Chinese Academy of Sciences, Beijing, 100190, China

    Xuejun Xu

  4. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

    Shangyou Zhang

Authors
  1. Liwei Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xuejun Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shangyou Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors made equally contributions.

Corresponding author

Correspondence to Shangyou Zhang.

Ethics declarations

Conflict of interest

There is no potential conflict of interest.

Ethical Standards

The submitted work is original and is not published elsewhere in any form or language.

Ethical approval

This article does not contain any studies involving animals. This article does not contain any studies involving human participants.

Informed consent

This research does not have any human participant.

Additional information

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

Xu, L., Xu, X. & Zhang, S. A Modified Rotated-\(Q_1\) Finite Element for the Stokes Equations on Quadrilateral and Hexahedral Meshes. J Sci Comput 99, 11 (2024). https://doi.org/10.1007/s10915-024-02477-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-024-02477-6

Keywords

Mathematics Subject Classification

Search

Navigation