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Computational Methods in Applied Mathematics

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Positivity Preserving Gradient Approximation with Linear Finite Elements

Andreas VeeserORCID iD: http://orcid.org/0000-0002-2152-2911
Published Online: 2018-06-30 | DOI: https://doi.org/10.1515/cmam-2018-0017


Preserving positivity precludes that linear operators onto continuous piecewise affine functions provide near best approximations of gradients. Linear interpolation thus does not capture the approximation properties of positive continuous piecewise affine functions. To remedy, we assign nodal values in a nonlinear fashion such that their global best error is equivalent to a suitable sum of local best errors with positive affine functions. As one of the applications of this equivalence, we consider the linear finite element solution to the elliptic obstacle problem and derive that its error is bounded in terms of these local best errors.

Keywords: Approximation of Gradients; Courant Elements; Best Error Decompositions; Positivity; Obstacles; A Priori Error Bounds

MSC 2010: 65N30; 41A29; 41A05; 41A36; 65N15

Dedicated to Amiya Kumar Pani on the occasion of his 60th birthday


  • [1]

    P. Binev and R. DeVore, Fast computation in adaptive tree approximation, Numer. Math. 97 (2004), no. 2, 193–217. CrossrefGoogle Scholar

  • [2]

    S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Texts Appl. Math. 15, Springer, New York, 2008. Google Scholar

  • [3]

    F. Brezzi, W. W. Hager and P.-A. Raviart, Error estimates for the finite element solution of variational inequalities, Numer. Math. 28 (1977), no. 4, 431–443. CrossrefGoogle Scholar

  • [4]

    Z. Chen and R. H. Nochetto, Residual type a posteriori error estimates for elliptic obstacle problems, Numer. Math. 84 (2000), no. 4, 527–548. CrossrefGoogle Scholar

  • [5]

    P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Stud. Math. Appl. 4, North-Holland Publishing, Amsterdam, 1978, Google Scholar

  • [6]

    P. G. Ciarlet, Basic error estimates for elliptic problems, Handbook of Numerical Analysis. Vol. II, North-Holland, Amsterdam (1991), 17–352. Google Scholar

  • [7]

    R. S. Falk, Error estimates for the approximation of a class of variational inequalities, Math. Comput. 28 (1974), 963–971. CrossrefGoogle Scholar

  • [8]

    F. Fierro and A. Veeser, A posteriori error estimators for regularized total variation of characteristic functions, SIAM J. Numer. Anal. 41 (2003), 2032–2055. CrossrefGoogle Scholar

  • [9]

    D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer, Berlin, 2001. Google Scholar

  • [10]

    R. Nochetto, K. Siebert and A. Veeser, Fully localized a posteriori error estimators and barrier sets for contact problems, SIAM J. Numer. Anal. 42 (2005), 2118–2135. CrossrefGoogle Scholar

  • [11]

    R. H. Nochetto and L. B. Wahlbin, Positivity preserving finite element approximation, Math. Comp. 71 (2002), no. 240, 1405–1419. Google Scholar

  • [12]

    L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. CrossrefGoogle Scholar

  • [13]

    A. Veeser, Approximating gradients with continuous piecewise polynomial functions, Found. Comput. Math. 16 (2016), no. 3, 723–750. CrossrefWeb of ScienceGoogle Scholar

  • [14]

    A. Veeser and R. Verfürth, Explicit upper bounds for dual norms of residuals, SIAM J. Numer. Anal. 47 (2009), no. 3, 2387–2405. Web of ScienceCrossrefGoogle Scholar

About the article

Received: 2018-12-18

Revised: 2018-04-30

Accepted: 2018-06-07

Published Online: 2018-06-30

The author is a member of the INdAM research group GNCS, the support of which is gratefully acknowledged.

Citation Information: Computational Methods in Applied Mathematics, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2018-0017.

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