# Localization of the $W^{-1,q}$ norm for local a posteriori efficiency

Abstract : This paper gives a direct proof of localization of dual norms of bounded linear functionals on the Sobolev space $W^{1,p}_0(\Omega)$. The basic condition is that the functional in question vanishes over locally supported test functions from $W^{1,p}_0(\Omega)$ which form a partition of unity in $\Omega$, apart from close to the boundary $\partial \Omega$. We also study how to weaken this condition. The results allow in particular to establish local efficiency and robustness of a posteriori estimates for nonlinear partial differential equations in divergence form, including the case of inexact solvers.
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https://hal.inria.fr/hal-01332481
Contributor : Martin Vohralik <>
Submitted on : Sunday, January 15, 2017 - 10:08:10 PM
Last modification on : Tuesday, October 16, 2018 - 12:45:06 PM

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• HAL Id : hal-01332481, version 2

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Jan Blechta, Josef Málek, Martin Vohralík. Localization of the $W^{-1,q}$ norm for local a posteriori efficiency. 2016. ⟨hal-01332481v2⟩

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