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Calcolo

, Volume 54, Issue 3, pp 1009–1025 | Cite as

Discrete p-robust \(\varvec{H}({{\mathrm{div}}})\)-liftings and a posteriori estimates for elliptic problems with \(H^{-1}\) source terms

  • Alexandre Ern
  • Iain SmearsEmail author
  • Martin Vohralík
Article

Abstract

We establish the existence of liftings into discrete subspaces of \(\varvec{H}({{\mathrm{div}}})\) of piecewise polynomial data on locally refined simplicial partitions of polygonal/polyhedral domains. Our liftings are robust with respect to the polynomial degree. This result has important applications in the a posteriori error analysis of parabolic problems, where it permits the removal of so-called transition conditions that link two consecutive meshes. It can also be used in the a posteriori error analysis of elliptic problems, where it allows the treatment of meshes with arbitrary numbers of hanging nodes between elements. We present a constructive proof based on the a posteriori error analysis of an auxiliary elliptic problem with \(H^{-1}\) source terms, thereby yielding results of independent interest. In particular, for such problems, we obtain guaranteed upper bounds on the error along with polynomial-degree robust local efficiency of the estimators.

Supplementary material

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Copyright information

© Springer-Verlag Italia 2017

Authors and Affiliations

  1. 1.Université Paris-Est, CERMICS (ENPC)Marne-la-Vallée 2France
  2. 2.Inria ParisParisFrance

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