Oscillation in a posteriori error analysis

A posteriori error estimators are a key tool for the quality assessment of given finite element approximations to an unknown PDE solution as well as for the application of adaptive techniques. Typically, the estimators are equivalent to the error up to an additive term, the so called oscillation. It is a common believe that this is the price for the `computability' of the estimator and that the oscillation is of higher order than the error. Cohen, DeVore, and Nochetto [CoDeNo:2012], however, presented an example, where the error vanishes with the generic optimal rate, but the oscillation does not. Interestingly, in this example, the local $H^{-1}$-norms are assumed to be computed exactly and thus the computability of the estimator cannot be the reason for the asymptotic overestimation. In particular, this proves both believes wrong in general. In this talk, we present a new approach to posteriori error analysis, where the oscillation is dominated by the error. The crucial step is a new splitting of the data into oscillation and oscillation free data. Moreover, the estimator is computable if the discrete linear system can essentially be assembled exactly.