Oscillation in a posteriori error analysis
A talk in the Keine Reihe series by
Christian Kreuzer
| Abstract: | 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.
[CoDeNo:2012] A. Cohen, R. DeVore, and R. H. Nochetto,
Convergence Rates of AFEM with H −1 Data,
Found Comput Math12 (2012):671–718
Joint work with Andreas Veeser (Università degli Studi di Milano, Italy). |