Skip to main content
SpringerLink
Log in

Order two superconvergence of the CDG finite elements for non-self adjoint and indefinite elliptic equations

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

Abstract

A conforming discontinuous Galerkin (CDG) finite element method is designed for solving second order non-self adjoint and indefinite elliptic equations. Unlike other discontinuous Galerkin (DG) methods, the numerical trace on the edge/triangle between two elements is not the average of two discontinuous \(P_k\) functions, but a lifted \(P_{k+2}\) function from four (eight in 3D) nearby \(P_k\) functions. While all existing DG methods have the optimal order of convergence, this CDG method has a superconvergence of order two above the optimal order when solving general second order elliptic equations. Due to the superconvergence, a post-process lifts a \(P_k\) CDG solution to a quasi-optimal \(P_{k+2}\) solution on each element. Numerical tests in 2D and 3D are provided confirming 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

Similar content being viewed by others

References

  1. AL-Taweel, A., Wang, X., Ye, X., Zhang, S.: A stabilizer free weak Galerkin method with supercloseness of order two. Numer. Meth. PDE. 37, 1012–1029 (2021)

    Article  MathSciNet  Google Scholar 

  2. Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  3. Cui, M., Zhang, S.: On the uniform convergence of the weak Galerkin finite element method for a singularly-perturbed biharmonic equation. J. Sci. Comput. 82, 5–15 (2020)

    Article  MathSciNet  Google Scholar 

  4. Feng, Y., Liu, Y., Wang, R., Zhang, S.: A conforming discontinuous Galerkin finite element method on rectangular partitions. Electron. Res. Arch. 29, 2375–2389 (2021)

    Article  MathSciNet  Google Scholar 

  5. Lin, R., Ye, X., Zhang, S., Zhu, P.: A weak Galerkin finite element method for singularly perturbed convection-diffusion-reaction problems. SIAM J. Num. Anal. 56, 1482–1497 (2018)

    Article  MathSciNet  Google Scholar 

  6. Mu, L., Ye, X., Zhang, S.: A stabilizer free, pressure robust and superconvergence weak Galerkin finite element method for the Stokes Equations on polytopal mesh. SIAM J. Sci. Comput. 43, A2614–A2637 (2021)

    Article  MathSciNet  Google Scholar 

  7. Wang, J., Zhai, Q., Zhang, R., Zhang, S.: A weak Galerkin finite element scheme for the Cahn-Hilliard equation. Math. Comp. 88, 211–235 (2019)

    Article  MathSciNet  Google Scholar 

  8. Ye, X., Zhang, S.: A stabilizer free weak Galerkin method for the biharmonic equation on polytopal meshes. SIAM J. Numer. Anal. 58, 2572–2588 (2020)

    Article  MathSciNet  Google Scholar 

  9. Ye, X., Zhang, S.: A conforming discontinuous Galerkin finite element method. Int. J. Numer. Anal. Model. 17(1), 110–117 (2020)

    MathSciNet  Google Scholar 

  10. Ye, X., Zhang, S.: A conforming discontinuous Galerkin finite element method: part II. Int. J Numer. Anal. Model. 17, 281–296 (2020)

    MathSciNet  Google Scholar 

  11. Ye, X., Zhang, S.: A conforming discontinuous Galerkin finite element method: part III. Int. J. Numer. Anal. Model. 17(6), 794–805 (2020)

    MathSciNet  Google Scholar 

  12. Ye, X., Zhang, S.: A conforming discontinuous Galerkin finite element method for the Stokes problem on polytopal meshes. Internat. J. Numer. Methods Fluids. 93(6), 1913–1928 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  13. Ye, X., Zhang, S.: A C0-conforming DG finite element method for biharmonic equations on triangle/tetrahedron. J. Numer. Math. 30(3), 163–172 (2021)

    Article  Google Scholar 

  14. Ye, X., Zhang, S.: A weak divergence CDG method for the biharmonic equation on triangular and tetrahedral meshes. Appl. Numer. Math. 178, 155–165 (2022)

    Article  MathSciNet  Google Scholar 

  15. Ye, X., Zhang, S.: Achieving superconvergence by one-dimensional discontinuous finite elements: the CDG method. East Asian J. Appl. Math. 12(4), 781–790 (2022)

    Article  MathSciNet  Google Scholar 

  16. Ye, X., Zhang, S.: Order two superconvergence of the CDG finite elements on triangular and tetrahedral meshes. CSIAM Trans. Appl. Math. 4(2), 256–274 (2023)

    Article  MathSciNet  Google Scholar 

  17. Zhu, P., Xie, S.: Superconvergent weak Galerkin methods for non-self adjoint and indefinite elliptic problems. Appl. Numer. Math. 172, 300–314 (2022)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Xiu Ye

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

    Shangyou Zhang

Authors
  1. Xiu Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. 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

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

Ye, X., Zhang, S. Order two superconvergence of the CDG finite elements for non-self adjoint and indefinite elliptic equations. Adv Comput Math 50, 2 (2024). https://doi.org/10.1007/s10444-023-10100-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10444-023-10100-9

Keywords

Mathematics Subject Classification (2010)

Search

Navigation