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

Alexandre Ern (1, 2) , Iain Smears (2, 1) , Martin Vohralík (2, 1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms

We establish the existence of liftings into discrete subspaces of H(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 a the 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.

Mots clés: stability of mixed methods; Hdiv liftings; right-inverse of divergence; a posteriori error analysis; polynomial degree robustness

Domaines: Analyse numérique [math.NA]

Alexandre Ern, Iain Smears, Martin Vohralík. Discrete $p$-robust H(div)-liftings and a posteriori estimates for elliptic problems with $H^{-1}$ source terms. Calcolo, 2017, 54 (3), pp.1009-1025. ⟨10.1007/s10092-017-0217-4⟩. ⟨hal-01377007⟩