, 56:44 | Cite as

Virtual enriching operators

  • Susanne C. BrennerEmail author
  • Li-Yeng Sung


We construct bounded linear operators that map \(H^1\) conforming Lagrange finite element spaces to \(H^2\) conforming virtual element spaces in two and three dimensions. These operators are useful for the analysis of nonstandard finite element methods.


Virtual elements Fourth order elliptic boundary value problems Enriching operator 

Mathematics Subject Classification

65N30 35J40 


Supplementary material


  1. 1.
    Adams, R., Fournier, J.: Sobolev Spaces, 2nd edn. Academic Press, Amsterdam (2003)zbMATHGoogle Scholar
  2. 2.
    Beirão da Veiga, L., Dassi, F., Russo, A.: A \(C^1\) virtual element method on polyhedral meshes. arXiv:1808.01105v2 [math.NA] (to appear in Comput. Math. Appl.)
  3. 3.
    Bramble, J., Hilbert, S.: Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation. SIAM J. Numer. Anal. 7, 113–124 (1970)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brenner, S.: Two-level additive Schwarz preconditioners for nonconforming finite element methods. Math. Comput. 65, 897–921 (1996)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Brenner, S.: Convergence of nonconforming multigrid methods without full elliptic regularity. Math. Comput. 68, 25–53 (1999) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Brenner, S.: \(C^0\) interior penalty methods. In: Blowey, J., Jensen, M. (eds.) Frontiers in Numerical Analysis-Durham 2010, Lecture Notes in Computational Science and Engineering, vol. 85, pp. 79–147. Springer, Berlin (2012)CrossRefGoogle Scholar
  7. 7.
    Brenner, S., Gedicke, J., Sung, L.Y., Zhang, Y.: An a posteriori analysis of \(C^0\) interior penalty methods for the obstacle problem of clamped Kirchhoff plates. SIAM J. Numer. Anal. 55, 87–108 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Brenner, S., Gudi, T., Sung, L.Y.: An a posteriori error estimator for a quadratic \({C^0}\) interior penalty method for the biharmonic problem. IMA J. Numer. Anal. 30, 777–798 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)CrossRefGoogle Scholar
  10. 10.
    Brenner, S., Sung, L.Y.: A new convergence analysis of finite element methods for elliptic distributed optimal control problems with pointwise state constraints. SIAM J. Control Optim. 55, 2289–2304 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Brenner, S., Wang, K.: Two-level additive Schwarz preconditioners for \(C^0\) interior penalty methods. Numer. Math. 102, 231–255 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Brezzi, F., Marini, L.: Virtual element methods for plate bending problems. Comput. Methods Appl. Mech. Eng. 253, 455–462 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chinosi, C., Marini, L.: Virtual element method for fourth order problems: \(L^2\)-estimates. Comput. Math. Appl. 72, 1959–1967 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ciarlet, P.: Sur l’élément de Clough et Tocher. RAIRO Anal. Numér. 8, 19–27 (1974)zbMATHGoogle Scholar
  15. 15.
    Ciarlet, P.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)zbMATHGoogle Scholar
  16. 16.
    Clough, R., Tocher, J.: Finite element stiffness matrices for analysis of plate bending. In: Proceedings of Conference on Matrix Methods in Structural Mechanics, pp. 515–545. Wright-Patterson Air Force Base (1965)Google Scholar
  17. 17.
    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. R.A.I.R.O. Modél. Math. Anal. Numér. 13, 227–255 (1979)zbMATHGoogle Scholar
  18. 18.
    Dupont, T., Scott, R.: Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34, 441–463 (1980)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Georgoulis, E., Houston, P., Virtanen, J.: An a posteriori error indicator for discontinuous Galerkin approximations of fourth-order elliptic problems. IMA J. Numer. Anal. 31, 281–298 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains, Classics in Applied Mathematics, vol. 69. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2011)CrossRefGoogle Scholar
  21. 21.
    Lax, P.: Functional Analysis. Wiley-Interscience, New York (2002)zbMATHGoogle Scholar
  22. 22.
    Nečas, J.: Direct Methods in the Theory of Elliptic Equations. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  23. 23.
    Neilan, M., Wu, M.: Discrete Miranda–Talenti estimates and applications to linear and nonlinear PDEs. J. Comput. Appl. Math. 356, 358–376 (2019)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Peano, A.: Hierarchies of conforming finite elements for plane elasticity and plate bending. Comput. Math. Appl. 2, 211–224 (1976)CrossRefGoogle Scholar
  25. 25.
    Percell, P.: On cubic and quartic Clough–Tocher finite elements. SIAM J. Numer. Anal. 13, 100–103 (1976)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987)CrossRefGoogle Scholar
  27. 27.
    Worsey, A., Farin, G.: An \(n\)-dimensional Clough–Tocher interpolant. Constr. Approx. 3, 99–110 (1987)MathSciNetCrossRefGoogle Scholar

Copyright information

© Istituto di Informatica e Telematica (IIT) 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Center for Computation and TechnologyLouisiana State UniversityBaton RougeUSA

Personalised recommendations