A safeguarded dual weighted residual method

The dual weighted residual (DWR) method yields reliable a posteriori error bounds for linear output functionals provided that the error incurred by the numerical approximation of the dual solution is negligible. In that case, its performance is generally superior than that of global 'energy norm' error estimators which are 'unconditionally' reliable. We present a simple numerical example for which neglecting the approximation error leads to severe underestimation of the functional error, thus showing that the DWR method may be unreliable. We propose a remedy that preserves the original performance, namely a DWR method safeguarded by additional asymptotically higher order a posteriori terms. In particular, the enhanced estimator is unconditionally reliable and asymptotically coincides with the original DWR method. These properties are illustrated via the aforementioned example.