Numerische Mathematik

, Volume 143, Issue 4, pp 923–942 | Cite as

Approximation in FEM, DG and IGA: a theoretical comparison

  • Andrea Bressan
  • Espen SandeEmail author


In this paper we compare the approximation properties of degree p spline spaces with different numbers of continuous derivatives. We prove that, for a given space dimension, \(\mathcal {C}^{p-1}\) splines provide better a priori error bounds for the approximation of functions in \(H^{p+1}(0,1)\). Our result holds for all practically interesting cases when comparing \(\mathcal {C}^{p-1}\) splines with \(\mathcal {C}^{-1}\) (discontinuous) splines. When comparing \(\mathcal {C}^{p-1}\) splines with \(\mathcal {C}^{0}\) splines our proof covers almost all cases for \(p\ge 3\), but we can not conclude anything for \(p=2\). The results are generalized to the approximation of functions in \(H^{q+1}(0,1)\) for \(q<p\), to broken Sobolev spaces and to tensor product spaces.

Mathematics Subject Classification

41A15 41A44 65D07 74S05 



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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PaviaPaviaItaly
  2. 2.Department of MathematicsUniversity of Rome Tor VergataRomeItaly

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