Journal of Scientific Computing

, Volume 78, Issue 3, pp 1405–1437 | Cite as

The Hessian Discretisation Method for Fourth Order Linear Elliptic Equations

  • Jérôme Droniou
  • Bishnu P. Lamichhane
  • Devika ShylajaEmail author


In this paper, we propose a unified framework, the Hessian discretisation method (HDM), which is based on four discrete elements (called altogether a Hessian discretisation) and a few intrinsic indicators of accuracy, independent of the considered model. An error estimate is obtained, using only these intrinsic indicators, when the HDM framework is applied to linear fourth order problems. It is shown that HDM encompasses a large number of numerical methods for fourth order elliptic problems: finite element methods (conforming and non-conforming) as well as finite volume methods. We also use the HDM to design a novel method, based on conforming \(\mathbb {P}_1\) finite element space and gradient recovery operators. Results of numerical experiments are presented for this novel scheme and for a finite volume scheme.


Fourth order elliptic equations Numerical schemes Error estimates Hessian discretisation method Hessian schemes Finite element method Finite volume method Gradient recovery method 


  1. 1.
    Balasundaram, S., Bhattacharyya, P.K.: A mixed finite element method for fourth order elliptic equations with variable coefficients. Comput. Math. Appl. 10(3), 245–256 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bhattacharyya, P.K.: Mixed finite element methods for fourth order elliptic problems with variable coefficients. In: Brebbia, C.A. (ed.) Variational Methods in Engineering (Southampton, 1985), pp. 2.3–2.12. Springer, Berlin (1985)Google Scholar
  3. 3.
    Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994)zbMATHCrossRefGoogle Scholar
  4. 4.
    Chen, H., Guo, H., Zhang, Z., Zou, Q.: A \({C}^0\) linear finite element method for two fourth-order eigenvalue problems. IMA J. Numer. Anal. (2016). CrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, J., Wang, D., Du, Q.: Linear finite element super-convergence on simplicial meshes. Math. Comput. 83, 2161–2185 (2014)zbMATHCrossRefGoogle Scholar
  6. 6.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Elsevier, New York (1978)zbMATHGoogle Scholar
  7. 7.
    Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Mathématiques and Applications (Berlin) [Mathematics and Applications], vol. 69. Springer, Heidelberg (2012)Google Scholar
  8. 8.
    Douglas Jr., J., Dupont, T., Percell, P., Scott, R.: A family of \(C^{1}\) finite elements with optimal approximation properties for various Galerkin methods for 2nd and 4th order problems. RAIRO Anal. Numér. 13(3), 227–255 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Droniou, J., Eymard, R.: The asymmetric gradient discretisation method. In: Finite Volumes for Complex Applications VIII—Methods and Theoretical Aspects, volume 199 of Springer Proceedings of Mathematics and Statistics, pp. 311–319. Springer, Cham (2017)Google Scholar
  10. 10.
    Droniou, J., Eymard, R., Gallouët, T., Guichard, C., Herbin, R.: The Gradient Discretisation Method, Mathematics and Applications, 82. Springer, Berlin (2018).
  11. 11.
    Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Ciarlet, P.G., Lions, J.-L. (eds.) Techniques of Scientific Computing. Part III, Handbook of Numerical Analysis, VII, pp. 713–1020. North-Holland, Amsterdam (2000)Google Scholar
  12. 12.
    Eymard, R., Gallouët, T., Herbin, R., Linke, A.: Finite volume schemes for the biharmonic problem on general meshes. Math. Comput. 81(280), 2019–2048 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Eymard, R., Herbin, R.: Approximation of the biharmonic problem using piecewise linear finite elements. C. R. Math. Acad. Sci. Paris 348(23–24), 1283–1286 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Falk, R.S.: Approximation of the biharmonic equation by a mixed finite element method. SIAM J. Numer. Anal. 15(3), 556–567 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Herbin, R., Hubert, F.: Benchmark on discretization schemes for anisotropic diffusion problems on general grids. In: Eymard, R. (ed.) Finite Volumes for Complex Applications V, pp. 659–692. ISTE, London (2008)zbMATHGoogle Scholar
  16. 16.
    Kim, C., Lazarov, R., Pasciak, J., Vassilevski, P.: Multiplier spaces for the mortar finite element method in three dimensions. SIAM J. Numer. Anal. 39, 519–538 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Lamichhane, B.: Higher Order Mortar Finite Elements with Dual Lagrange Multiplier Spaces and Applications. Ph.D. thesis, Universität Stuttgart (2006)Google Scholar
  18. 18.
    Lamichhane, B.: A stabilized mixed finite element method for the biharmonic equation based on biorthogonal systems. J. Comput. Appl. Math. 235, 5188–5197 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Lamichhane, B., Stevenson, R., Wohlmuth, B.: Higher order mortar finite element methods in 3D with dual Lagrange multiplier bases. Numer. Math. 102, 93–121 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Lamichhane, B.P.: A stabilized mixed finite element method for the biharmonic equation based on biorthogonal systems. J. Comput. Appl. Math. 235(17), 5188–5197 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Lamichhane, B.P.: A finite element method for a biharmonic equation based on gradient recovery operators. BIT 54(2), 469–484 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Lascaux, P., Lesaint, P.: Some nonconforming finite elements for the plate bending problem. Rev. Fr. Autom. Inform. Rech. Oper. Rouge Anal. Numér. 9(R–1), 9–53 (1975)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Li, J.: Full-order convergence of a mixed finite element method for fourth-order elliptic equations. J. Math. Anal. Appl. 230(2), 329–349 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Li, Y., An, R., Li, K.: Some optimal error estimates of biharmonic problem using conforming finite element. Appl. Math. Comput. 194(2), 298–308 (2007)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Nataraj, N., Bhattacharyya, P.K., Balasundaram, S., Gopalsamy, S.: On a mixed-hybrid finite element method for anisotropic plate bending problems. Int. J. Numer. Methods Eng. 39(23), 4063–4089 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Percell, P.: On cubic and quartic Clough–Tocher finite elements. SIAM J. Numer. Anal. 13(1), 100–103 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Powell, M.J.D., Sabin, M.A.: Piecewise quadratic approximations on triangles. ACM Trans. Math. Softw. 3(4), 316–325 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Scapolla, T.: A mixed finite element method for the biharmonic problem. RAIRO Anal. Numér. 14(1), 55–79 (1980)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Xie, Z.H.: Error estimate of nonconforming finite element approximation for a fourth order elliptic variational inequality. Northeast. Math. J. 8(3), 329–336 (1992)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Xu, J., Zhang, Z.: Analysis of recovery type a posteriori error estimators for mildly structured grids. Math. Comput. 73, 1139–1152 (2004)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesMonash UniversityClaytonAustralia
  2. 2.School of Mathematical and Physical SciencesUniversity of NewcastleCallaghanAustralia
  3. 3.IITB-Monash Research AcademyIndian Institute of Technology BombayPowaiIndia

Personalised recommendations