Best Error Localizations for Piecewise Polynomial Approximation of Gradients, Functions and Functionals

  • Andreas VeeserEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)


We consider the approximation of (generalized) functions with continuous piecewise polynomials or with piecewise polynomials that are allowed to be discontinuous. Best error localization then means that the best error in the whole domain is equivalent to an appropriate accumulation of best errors in small domains, e.g., in mesh elements. We review and compare such best error localizations in the three cases of the Sobolev-Hilbert triplet \((H^1_0,L^2,H^{-1})\).



The support by the GNCS, part of the Italian INdAM, is gratefully acknowledged.


  1. 1.
    P. Binev, R. DeVore, Fast computation in adaptive tree approximation. Numer. Math. 97(2), 193–217 (2004)MathSciNetCrossRefGoogle Scholar
  2. 2.
    P. Binev, W. Dahmen, R. DeVore, Adaptive finite element methods with convergence rates. Numer. Math. 97(2), 219–268 (2004)MathSciNetCrossRefGoogle Scholar
  3. 3.
    F. Camacho, A. Demlow, L2 and pointwise a posteriori error estimates for fem for elliptic pdes on surfaces. IMA J. Numer. Anal. 35(3), 1199–1227 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    T. Dupont, R. Scott, Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34(150), 441–463 (1980)MathSciNetCrossRefGoogle Scholar
  5. 5.
    L.R. Scott, S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)MathSciNetCrossRefGoogle Scholar
  6. 6.
    F. Tantardini, A. Veeser, The L 2-projection and quasi-optimality of Galerkin methods for parabolic equations. SIAM J. Numer. Anal. 54(1), 317–340 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    F. Tantardini, A. Veeser, Quasi-optimality constants for parabolic galerkin approximation in space, in Numerical Mathematics and Advanced Applications ENUMATH 2015, ed. by B. Karasözen et al. (Springer International Publishing, Cham, 2016), pp. 105–113CrossRefGoogle Scholar
  8. 8.
    F. Tantardini, A. Veeser, R. Verfürth, H −1-approximation with piecewise polynomials (in preparation)Google Scholar
  9. 9.
    F. Tantardini, A. Veeser, R. Verfürth, Robust localization of the best error with finite elements in the reaction-diffusion norm. Constr. Approx. 42(2), 313–347 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    A. Veeser, Approximating gradients with continuous piecewise polynomial functions. Found. Comput. Math. 16(3), 723–750 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    A. Veeser, P. Zanotti, Quasi-optimal nonconforming methods for symmetric elliptic problems. III – DG and other interior penalty methods. SIAM J. Numer. Anal. (accepted for publication)Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Università degli Studi di MilanoDipartimento di Matematica F. EnriquesMilanoItaly

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