Variational localizations of the dual weighted residual estimator

Abstract

The dual weighted residual method (DWR) and its localization for mesh adaptivity applied to elliptic partial differential equations are investigated. The contribution of this paper is twofold: first, we introduce a novel localization technique based on the introduction of a partition of unity. This new technique is very easy to apply, as neither strong residuals nor jumps over element edges are required. Second, we compare and analyze (theoretically and numerically) different localization techniques used for mesh adaptivity with respect to their effectivity. Here, we focus on localizations in variational formulations that do not require the evaluation of the corresponding differential operator in the classical strong formulation. In our mathematical analysis, we show for different localization techniques (established methods and our new approach) that the local error indicators used for mesh adaptivity converge with proper order in the error functional. Several numerical tests substantiate our theoretical investigations.