Discrete p-robust $$\varvec{H}({{\mathrm{div}}})$$-liftings and a posteriori estimates for elli ptic problems with $$H^{-1}$$H-1 source terms |
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Authors: | Alexandre Ern Iain Smears Martin Vohralík |
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Affiliation: | 1.Université Paris-Est, CERMICS (ENPC),Marne-la-Vallée 2,France;2.Inria Paris,Paris,France |
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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. |
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