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Journal Articles IMA Journal of Numerical Analysis Year : 2020

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

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Josef Málek
• Function : Author
Martin Vohralík

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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.

#### Domains

Mathematics [math] Numerical Analysis [math.NA]

### Dates and versions

hal-01332481 , version 1 (16-06-2016)
hal-01332481 , version 2 (15-01-2017)
hal-01332481 , version 3 (04-07-2018)

### Identifiers

• HAL Id : hal-01332481 , version 3
• DOI :

### Cite

Jan Blechta, Josef Málek, Martin Vohralík. Localization of the $W^{-1,q}$ norm for local a posteriori efficiency. IMA Journal of Numerical Analysis, 2020, 40 (2), pp.914-950. ⟨10.1093/imanum/drz002⟩. ⟨hal-01332481v3⟩

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