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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
Article
  • 173 Downloads

Abstract

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 

Notes

References

  1. 1.
    Bazilevs, Y., Beirão da Veiga, L., Cottrell, J.A., Hughes, T.J.R., Sangalli, G.: Isogeometric analysis: approximation, stability and error estimates for \(h\)-refined meshes. Math. Models Methods Appl. Sci. 16(7), 1031–1090 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Beirão da Veiga, L., Buffa, A., Rivas, J., Sangalli, G.: Some estimates for \(h\)\(p\)\(k\)-refinement in isogeometric analysis. Numer. Math. 118(2), 271–305 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Beirão da Veiga, L., Buffa, A., Sangalli, G., Vázquez, R.: Mathematical analysis of variational isogeometric methods. Acta Numer. 23, 157–287 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Evans, J.A., Bazilevs, Y., Babuška, I., Hughes, T.J.R.: \(n\)-Widths, sup–infs, and optimality ratios for the \(k\)-version of the isogeometric finite element method. Comput. Methods Appl. Mech. Eng. 198(21–26), 1726–1741 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Floater, M.S., Sande, E.: Optimal spline spaces of higher degree for \(L^2\) \(n\)-widths. J. Approx. Theory 216, 1–15 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Floater, M.S., Sande, E.: Optimal spline spaces for \(L^2\) \(n\)-width problems with boundary conditions. Constr. Approx. 50, 1–18 (2019)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194(39–41), 4135–4195 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kolmogorov, A.: Über die beste Annäherung von Funktionen einer gegebenen Funktionenklasse. Ann. Math. 37, 107–110 (1936)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Melkman, A.A., Micchelli, C.A.: Spline spaces are optimal for \(L^2\) \(n\)-width. Ill. J. Math. 22, 541–564 (1978)CrossRefGoogle Scholar
  10. 10.
    Pinkus, A.: \(n\)-Widths in Approximation Theory. Springer, Berlin (1985)zbMATHGoogle Scholar
  11. 11.
    Robbins, H.: A remark on stirling’s formula. Am. Math. Mon. 62(1), 26–29 (1955)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Takacs, S., Takacs, T.: Approximation error estimates and inverse inequalities for B-splines of maximum smoothness. Math. Models Methods Appl. Sci. 26(7), 1411–1445 (2016)MathSciNetCrossRefGoogle Scholar

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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