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

Abstract : 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.
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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, Springer Verlag, 2017, 54 (3), pp.1009-1025. ⟨https://link.springer.com/article/10.1007/s10092-017-0217-4⟩. ⟨10.1007/s10092-017-0217-4⟩. ⟨hal-01377007⟩

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