Skip to main content

Inexpensive guaranteed and efficient upper bounds on the algebraic error in finite element discretizations


We present new constructions of (approximate) H(div, Ω)-liftings of the algebraic residual leading to estimators of the algebraic error in h and p finite element discretizations of a model diffusion problem. The estimators provide guaranteed bounds without any uncomputable constants and they are globally efficient, similarly to some recent developments, but the cost of their construction is significantly reduced. We provide a set of numerical experiments to assess the performance of the new estimators.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11


  1. 1.

    Arioli, M., Georgoulis, E.H., Loghin, D.: Stopping criteria for adaptive finite element solvers. SIAM J. Sci. Comput. 35(3), A1537–A1559 (2013).

    MathSciNet  Article  Google Scholar 

  2. 2.

    Arioli, M., Loghin, D., Wathen, A.J.: Stopping criteria for iterations in finite element methods. Numer. Math. 99(3), 381–410 (2005).

    MathSciNet  Article  Google Scholar 

  3. 3.

    Arnold, D.N., Falk, R.S., Winther, R.: Preconditioning in H(div) and applications. Math. Comp. 66 (219), 957–984 (1997).

    MathSciNet  Article  Google Scholar 

  4. 4.

    Arnold, D.N., Falk, R.S., Winther, R.: Multigrid in H(div) and H(curl). Numer. Math. 85 (2), 197–217 (2000).

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bai, D., Brandt, A.: Local mesh refinement multilevel techniques. SIAM J. Sci. Statist. Comput. 8(2), 109–134 (1987).

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bank, R.E., Sherman, A.H.: An adaptive, multilevel method for elliptic boundary value problems. Computing 26(2), 91–105 (1981).

    MathSciNet  Article  Google Scholar 

  7. 7.

    Bank, R.E., Smith, R.K.: A posteriori error estimates based on hierarchical bases. SIAM J. Numer. Anal. 30(4), 921–935 (1993).

    MathSciNet  Article  Google Scholar 

  8. 8.

    Becker, R., Johnson, C., Rannacher, R.: Adaptive error control for multigrid finite element methods. Computing 55(4), 271–288 (1995).

    MathSciNet  Article  Google Scholar 

  9. 9.

    Blechta, J., Málek, J., Vohralík, M.: Localization of the W− 1, q norm for local a posteriori efficiency. IMA J. Numer. Anal. 40(2), 914–950 (2020).

    MathSciNet  Article  Google Scholar 

  10. 10.

    Boffi, D., Brezzi, F., Fortin, M.: Mixed finite element methods and applications. Springer Series in Computational Mathematics, vol. 44. Springer, Heidelberg (2013).

    Book  Google Scholar 

  11. 11.

    Braess, D.: Finite elements. Theory, fast solvers, and applications in elasticity theory, Translated from the German by Larry L. Schumaker, 3rd edn. Cambridge University Press, Cambridge (2007)

  12. 12.

    Brandt, A.: Multi-level adaptive solutions to boundary-value problems. Math. Comp. 31(138), 333–390 (1977)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Destuynder, P., Métivet, B.: Explicit error bounds in a conforming finite element method. Math. Comp. 68(228), 1379–1396 (1999).

    MathSciNet  Article  Google Scholar 

  14. 14.

    Ern, A., Smears, I., Vohralík, M.: Discrete p-robust H(div)-liftings and a posteriori estimates for elliptic problems with H− 1 source terms. Calcolo 54(3), 1009–1025 (2017).

    MathSciNet  Article  Google Scholar 

  15. 15.

    Ern, A., Vohralík, M.: Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs. SIAM J. Sci. Comput. 35(4), A1761–A1791 (2013).

    MathSciNet  Article  Google Scholar 

  16. 16.

    Ern, A., Vohralík, M.: Stable broken H1 and H(div) polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions. Math. Comp. 89 (322), 551–594 (2020).

    MathSciNet  Article  Google Scholar 

  17. 17.

    Golub, G.H., Meurant, G.: Matrices, moments and quadrature with applications. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, NJ (2010)

    MATH  Google Scholar 

  18. 18.

    Hackbusch, W.: Multigrid methods and applications. Springer Series in Computational Mathematics, vol. 4. Springer-Verlag, Berlin (1985)

    Google Scholar 

  19. 19.

    Hecht, F.: New development in FreeFem++. J. Numer. Math. 20(3-4), 251–265 (2012).

    MathSciNet  Article  Google Scholar 

  20. 20.

    Hecht, F., Pironneau, O., Morice, J., Le Hyaric, A., Ohtsuka, K.: FreeFem++. Tech. rep., Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris. (2012)

  21. 21.

    Janssen, B., Kanschat, G.: Adaptive multilevel methods with local smoothing for H1- and Hcurl-conforming high order finite element methods. SIAM J. Sci. Comput. 33(4), 2095–2114 (2011).

    MathSciNet  Article  Google Scholar 

  22. 22.

    Jiránek, P., Strakoš, Z., Vohralík, M.: A posteriori error estimates including algebraic error and stopping criteria for iterative solvers. SIAM J. Sci. Comput. 32(3), 1567–1590 (2010).

    MathSciNet  Article  Google Scholar 

  23. 23.

    Luce, R., Wohlmuth, B.I.: A local a posteriori error estimator based on equilibrated fluxes. SIAM J. Numer. Anal. 42(4), 1394–1414 (2004).

    MathSciNet  Article  Google Scholar 

  24. 24.

    Meidner, D., Rannacher, R., Vihharev, J.: Goal-oriented error control of the iterative solution of finite element equations. J. Numer. Math. 17 (2), 143–172 (2009).

    MathSciNet  Article  Google Scholar 

  25. 25.

    Meurant, G., Tichý, P.: Approximating the extreme Ritz values and upper bounds for the A-norm of the error in CG. Numer. Algoritm. 82(3), 937–968 (2019).

    MathSciNet  Article  Google Scholar 

  26. 26.

    Miraçi, A., Papež, J., Vohralík, M.: A multilevel algebraic error estimator and the corresponding iterative solver with p-robust behavior. SIAM J. Numer. Anal. 58(5), 2856–2884 (2020).

    MathSciNet  Article  Google Scholar 

  27. 27.

    Oswald, P.: Multilevel finite element approximation. Teubner Skripten zur Numerik. [Teubner Scripts on Numerical Mathematics]. B. G. Teubner, Stuttgart. Theory and applications. (1994)

  28. 28.

    Papež, J., Rüde, U., Vohralík, M., Wohlmuth, B.: Sharp algebraic and total a posteriori error bounds for h and p finite elements via a multilevel approach. Recovering mass balance in any situation. Comput. Methods Appl. Mech. Engrg. 371, 113243 (2020).

    MathSciNet  Article  Google Scholar 

  29. 29.

    Papež, J., Strakoš, Z., Vohralík, M.: Estimating and localizing the algebraic and total numerical errors using flux reconstructions. Numer. Math. 138(3), 681–721 (2018).

    MathSciNet  Article  Google Scholar 

  30. 30.

    Rüde, U.: Fully adaptive multigrid methods. SIAM J. Numer. Anal. 30(1), 230–248 (1993)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Rüde, U.: Mathematical and computational techniques for multilevel adaptive methods. Frontiers in Applied Mathematics, vol. 13. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1993).

    Google Scholar 

  32. 32.

    Rüde, U.: Error estimates based on stable splittings. In: Domain decomposition methods in scientific and engineering computing (University Park, PA, 1993), Contemp. Math. Amer. Math. Soc., Providence, RI, vol. 180, pp 111–118 (1994)

  33. 33.

    Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia, PA (2003)

    Book  Google Scholar 

  34. 34.

    Si, H.: TetGen, a Delaunay-based quality tetrahedral mesh generator. ACM Trans. Math. Softw. 41(2), Art. 11,36 (2015).

    MathSciNet  Article  Google Scholar 

  35. 35.

    Strakoš, Z., Tichý, P.: On error estimation in the conjugate gradient method and why it works in finite precision computations. Electron. Trans. Numer. Anal. 13, 56–80 (2002)

    MathSciNet  MATH  Google Scholar 

Download references


This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement no. 647134 GATIPOR). The work of J. Papež was supported by the Grant Agency of the Czech Republic under grant no. 20-01074S in the framework of RVO 67985840.

Author information



Corresponding author

Correspondence to Jan Papež.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Papež, J., Vohralík, M. Inexpensive guaranteed and efficient upper bounds on the algebraic error in finite element discretizations. Numer Algor (2021).

Download citation


  • Finite element method
  • Iterative algebraic solver
  • Algebraic error
  • A posteriori error estimate
  • Guaranteed upper bound
  • Hierarchical splitting

Mathematics Subject Classification (2010)

  • 65N15
  • 65N30
  • 76M10
  • 65N22
  • 65F10