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

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Abstract : This paper gives a direct proof of localization of dual norms of bounded linear functionals on the Sobolev space $W^{1,p}_0(\Omega)$, $1 \leq p \leq \infty$. 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 with respect to the exponent $p$ of a posteriori estimates for nonlinear partial differential equations in divergence form, including the case of inexact solvers. Numerical illustrations support the theory.
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Journal articles

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https://hal.inria.fr/hal-01332481
Contributor : Martin Vohralik <>
Submitted on : Wednesday, July 4, 2018 - 5:39:50 PM
Last modification on : Tuesday, May 7, 2019 - 1:37:53 AM

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Jan Blechta, Josef Málek, Martin Vohralík. Localization of the $W^{-1,q}$ norm for local a posteriori efficiency. IMA Journal of Numerical Analysis, Oxford University Press (OUP), 2019, ⟨10.1093/imanum/drz002⟩. ⟨hal-01332481v3⟩

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