Skip to main content
SpringerLink
Log in

A Mixed Finite-Element Method on Polytopal Mesh

  • Original Paper
  • Published:
Communications on Applied Mathematics and Computation Aims and scope Submit manuscript

Abstract

In this paper, we introduce new stable mixed finite elements of any order on polytopal mesh for solving second-order elliptic problem. We establish optimal order error estimates for velocity and super convergence for pressure. Numerical experiments are conducted for our mixed elements of different orders on 2D and 3D spaces that confirm the theory.

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

Similar content being viewed by others

References

  1. Arnold, D., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Model. Math. Anal. Numer. 19, 7–32 (1985)

    Article  MathSciNet  Google Scholar 

  2. Babuska, I.: The finite element method with Lagrange multipliers. Numer. Math. 20, 179–192 (1973)

    Article  Google Scholar 

  3. Boffi, D., Brezzi, F., Demkowicz, L., Duran, R., Falk, R., Fortin, M.: Mixed finite elements, compatibility conditions, and applications. In: Boffi, D., Gastaldi, L. (eds) Lecture Notes in Mathematics. Springer-Verlag, Berlin (2008)

    Google Scholar 

  4. Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol. 44. Springer, Heidelberg (2013)

  5. Brezzi, F., Douglas, J., Duran, R., Fortin, M.: Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51, 237–250 (1987)

    Article  MathSciNet  Google Scholar 

  6. Brezzi, F., Douglas, J., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47, 217–235 (1985)

    Article  MathSciNet  Google Scholar 

  7. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Elements. Springer-Verlag, New York (1991)

    Book  Google Scholar 

  8. Chen, W., Wang, Y.: Minimal degree \(H({\text{curl}})\) and \(H({\text{div}})\) conforming finite elements on polytopal meshes. Math. Comput. 86, 2053–2087 (2017)

    Article  MathSciNet  Google Scholar 

  9. 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. 47, 1319–1365 (2009)

    Article  MathSciNet  Google Scholar 

  10. Di Pietro, D., Ern, A.: Hybrid high-order methods for variable-diffusion problems on general meshes. Comptes Rendus Mathmatique 353, 31–34 (2015)

    Article  MathSciNet  Google Scholar 

  11. Kuznetsov, Y., Repin, S.: New mixed finite element method on polygonal and polyhedral meshes. Russian J. Numer. Anal. Math. Model. 18(3), 261–278 (2003)

    Article  MathSciNet  Google Scholar 

  12. Kuznetsov, Y., Repin, S.: Convergence analysis and error estimates for mixed finite element method on distorted meshes. J. Numer. Math. 13(1), 33–51 (2005)

    Article  MathSciNet  Google Scholar 

  13. Lipnikov, K., Manzini, G., Brezzi, F., Buffa, A.: The mimetic finite difference method for the 3D magnetostatic field problems on polyhedral meshes. J. Comput. Phys. 230, 305–328 (2011)

    Article  MathSciNet  Google Scholar 

  14. Raviart, P., Thomas, J.: A mixed finite element method for second order elliptic problems. In: Galligani, I., Magenes, E. (eds) Mathematical Aspects of the Finite Element Method, Lectures Notes in Math. vol. 606. Springer-Verlag, New York (1977)

  15. Talischi, C., Paulino, G., Pereira, A., Menezes, I.: Polygonal finite elements for topology optimization: a unifying paradigm. Int. J. Numer. Meth. Eng. 82, 671–698 (2010)

    Article  Google Scholar 

  16. Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L., Russo, A.: Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23, 119–214 (2013)

    MathSciNet  MATH  Google Scholar 

  17. Wang, J.: Mixed finite element methods. In: Cai, W., Shi, Z., Shu, C.-W., Xu, J. (eds) Numerical Methods in Scientific and Engineering Computing. Academic Press, USA

  18. Wang, J., Ye, X.: A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. 241, 103–115 (2013)

    Article  MathSciNet  Google Scholar 

  19. Wang, J., Ye, X.: A weak Galerkin mixed finite element method for second-order elliptic problems. Math. Comput. 83, 2101–2126 (2014)

    Article  MathSciNet  Google Scholar 

  20. Ye, X., Zhang, S.: A conforming DG method for the biharmonic equation on polytopal. arXiv:1907.10661

Download references

Acknowledgements

This research was supported in part by the National Science Foundation Grant DMS-1620016. This work was also supported in parts by HKSAR grant Q81Q and JRI of The Hong Kong Polytechnic University.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong, China

    Yanping Lin

  2. Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR, 72204, USA

    Xiu Ye

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

    Shangyou Zhang

Authors
  1. Yanping Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiu Ye
    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

Corresponding author

Correspondence to Shangyou Zhang.

Ethics declarations

Conflict of Interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lin, Y., Ye, X. & Zhang, S. A Mixed Finite-Element Method on Polytopal Mesh. Commun. Appl. Math. Comput. 4, 1374–1385 (2022). https://doi.org/10.1007/s42967-021-00180-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s42967-021-00180-z

Keywords

Mathematics Subject Classification

Search

Navigation