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

Jan Blechta (1) , Josef Málek (1) , Martin Vohralík (2, 3)

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

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.

Mots clés: local structure; residual; Sobolev space W^{1p}_0(Ω); functional; dual norm; finite element method; nonlinear partial differential equation; a posteriori error estimate

Domaines: Analyse numérique [math.NA]

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