Numerische Mathematik

, Volume 132, Issue 3, pp 519–539 | Cite as

The norm of a discretized gradient in \(\varvec{H({{\mathrm{div}}})^*}\) for a posteriori finite element error analysis

  • Carsten Carstensen
  • Daniel Peterseim
  • Andreas Schröder


This paper characterizes the norm of the residual of mixed schemes in their natural functional framework with fluxes or stresses in \(H({{\mathrm{div}}})\) and displacements in \(L^2\). Under some natural conditions on an associated Fortin interpolation operator, reliable and efficient error estimates are introduced that circumvent the duality technique and so do not suffer from reduced elliptic regularity for non-convex domains. For the Laplace, Stokes, and Lamé equations, this generalizes known estimators to non-convex domains and introduces new a posteriori error estimators.

Mathematics Subject Classification

65N30 65N15 65N12 


  1. 1.
    Alonso, A.: Error estimators for a mixed method. Numer. Math. 74(4), 385–395 (1996). doi: 10.1007/s002110050222 CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Arnold, D.N., Brezzi, F., Douglas Jr, J.: PEERS: a new mixed finite element for plane elasticity. Jpn. J. Appl. Math. 1(2), 347–367 (1984). doi: 10.1007/BF03167064 CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Arnold, D.N., Winther, R.: Mixed finite elements for elasticity. Numer. Math. 92(3), 401–419 (2002). doi: 10.1007/s002110100348 CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Berndt, M., Manteuffel, T.A., McCormick, S.F.: Local error estimates and adaptive refinement for first-order system least squares (FOSLS). Electron. Trans. Numer. Anal. 6, 35–43 (1997) (electronic) (special issue on multilevel methods, Copper Mountain, CO, 1997)Google Scholar
  5. 5.
    Bochev, P.B., Gunzburger, M.D.: Applied Mathematical Sciences. Least-squares finite element methods, vol. 166. Springer, New York (2009)Google Scholar
  6. 6.
    Braess, D.: Finite elements, 3rd edn. Cambridge University Press, Cambridge (2007). doi: 10.1017/CBO9780511618635 (Theory, fast solvers, and applications in elasticity theory, translated from the German by Larry L. Schumaker)
  7. 7.
    Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. In: Texts in Applied Mathematics, vol 15, 3rd edn. Springer, New York (2008). doi: 10.1007/978-0-387-75934-0
  8. 8.
    Brezzi, F., Fortin, M.: Springer Series in Computational Mathematics. Mixed and hybrid finite element methods, vol. 15. Springer, New York (1991)Google Scholar
  9. 9.
    Cai, Z., Lee, B., Wang, P.: Least-squares methods for incompressible Newtonian fluid flow: linear stationary problems. SIAM J. Numer. Anal. 42(2), 843–859 (2004) (electronic). doi: 10.1137/S0036142903422673
  10. 10.
    Carstensen, C.: A posteriori error estimate for the mixed finite element method. Math. Comput. 66(218), 465–476 (1997). doi: 10.1090/S0025-5718-97-00837-5 CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Carstensen, C.: A unifying theory of a posteriori finite element error control. Numer. Math. 100(4), 617–637 (2005). doi: 10.1007/s00211-004-0577-y CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Carstensen, C., Bahriawati, C.: Three matlab implementations of the lowest-order Raviart–Thomas MFEM with a posteriori error control. CMAM 5(4), 333–361 (2005)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Carstensen, C., Dolzmann, G.: A posteriori error estimates for mixed FEM in elasticity. Numer. Math. 81(2), 187–209 (1998). doi: 10.1007/s002110050389 CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Carstensen, C., Hu, J.: A unifying theory of a posteriori error control for nonconforming finite element methods. Numer. Math. 107(3), 473–502 (2007). doi: 10.1007/s00211-007-0068-z CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Carstensen, C., Kim, D., Park, E.J.: A priori and a posteriori pseudostress-velocity mixed finite element error analysis for the Stokes problem. SIAM J. Numer. Anal. 49, 2501–2523 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Demlow, A., Hirani, A.N.: A posteriori error estimates for finite element exterior calculus: the de Rham complex. Found. Comput. Math. 14(6), 1337–1371 (2014). doi: 10.1007/s10208-014-9203-2 CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Gatica, G.N., Márquez, A., Sánchez, M.A.: Analysis of a velocity–pressure–pseudostress formulation for the stationary Stokes equations. Comput. Methods Appl. Mech. Eng. 199(17–20), 1064–1079 (2010). doi: 10.1016/j.cma.2009.11.024 CrossRefzbMATHGoogle Scholar
  18. 18.
    Gatica, G.N., Oyarzúa, R., Sayas, F.J.: A residual-based a posteriori error estimator for a fully-mixed formulation of the Stokes–Darcy coupled problem. Comput. Methods Appl. Mech. Eng. 200(21–22), 1877–1891 (2011). doi: 10.1016/j.cma.2011.02.009 CrossRefzbMATHGoogle Scholar
  19. 19.
    Schöberl, J.: A posteriori error estimates for Maxwell equations. Math. Comput. 77(262), 633–649 (2008). doi: 10.1090/S0025-5718-07-02030-3 CrossRefzbMATHGoogle Scholar
  20. 20.
    Stenberg, R.: A family of mixed finite elements for the elasticity problem. Numer. Math. 53(5), 513–538 (1988). doi: 10.1007/BF01397550 CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Carsten Carstensen
    • 1
  • Daniel Peterseim
    • 2
  • Andreas Schröder
    • 3
  1. 1.Institut für MathematikHumboldt Universität zu BerlinBerlinGermany
  2. 2.Institut für Numerische Simulation, Universität BonnBonnGermany
  3. 3.Fachbereich MathematikUniversität SalzburgSalzburgAustria

Personalised recommendations