A Cubic H3-Nonconforming Finite Element

  • Jun HuEmail author
  • Shangyou Zhang
Original Paper


The lowest degree of polynomial for a finite element to solve a 2kth-order elliptic equation is k. The Morley element is such a finite element, of polynomial degree 2, for solving a fourth-order biharmonic equation. We design a cubic \(H^3\)-nonconforming macro-element on two-dimensional triangular grids, solving a sixth-order tri-harmonic equation. We also write down explicitly the 12 basis functions on each macro-element. A convergence theory is established and verified by numerical tests.


Nonconforming macro-element Minimum element Tri-harmonic equation 

Mathematics Subject Classification

65N30 73C02 


  1. 1.
    Alfeld, P., Sirvent, M.: The structure of multivariate superspline spaces of high degree. Math. Comput. 57, 299–308 (1991)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Argyris, J.H., Fried, I., Scharpf, D.W.: The TUBA family of plate elements for the matrix displacement method. Aeronaut. J. R. Aeronaut. Soc. 72, 514–517 (1968)Google Scholar
  3. 3.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods Texts in Applied Mathematics, vol. 15, 3rd edn. Springer, New York (2008)Google Scholar
  4. 4.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)zbMATHGoogle Scholar
  5. 5.
    Gudi, T.: A new error analysis for discontinuous finite element methods for linear elliptic problems. Math. Comput. 79, 2169–2189 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hu, J., Ma, R., Shi, Z.: A new a priori error estimate of nonconforming finite element methods. Sci. China Math. 57, 887–902 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hu, J., Huang, Y., Zhang, S.: The lowest order differentiable finite element on rectangular grids. SIAM J. Numer. Anal. 49, 1350–1368 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hu, J., Zhang, S.: The minimal conforming \(H^k\) finite element spaces on \(\mathbb{R}^n\) rectangular grids. Math. Comput. 84, 563–579 (2015)CrossRefGoogle Scholar
  9. 9.
    Hu, J., Zhang, S.: A canonical construction of \(H^m\)-nonconforming triangular finite elements. Ann. Appl. Math. 33, 266–288 (2017)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Mao, S.P., Shi, Z.C.: On the error bounds of nonconforming finite elements. Sci. China Math. 53, 2917–2926 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Powell, M.J.D., Sabin, M.A.: Piecewise quadratic approximations on triangles. ACM Trans. Math. Softw. 3–4, 316–325 (1977)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, 483–493 (1990)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Shi, Z., Wang, M.: Mathematical Theory of Some Nonstandard Finite Element Methods. Computational Mathematics in China, Contemp. Math., vol. 163, pp. 111–125. American Mathematical Society, Providence, RI (1994)Google Scholar
  14. 14.
    Wang, M., Xu, J.: Minimal finite element spaces for 2m-th-order partial differential equations in \(\mathbb{R}^{n}\). Math. Comput. 82, 25–43 (2013)CrossRefGoogle Scholar
  15. 15.
    Wu, S., Xu, J.: Nonconforming finite element spaces for 2m-th order partial differential equations on \(\mathbb{R}^{n}\) simplicial grids when \(m = n + 1\). Math. Comp. 88, 531–551 (2019)Google Scholar
  16. 16.
    Zhang, S.: A family of 3D continuously differentiable finite elements on tetrahedral grids. Appl. Numer. Math. 59, 219–233 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Shanghai University 2019

Authors and Affiliations

  1. 1.LMAM and School of Mathematical SciencesPeking UniversityBeijingChina
  2. 2.Department of Mathematical SciencesUniversity of DelawareNewarkUSA

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