Estimation of Higher Sobolev Norm from Lower Order Approximation

Abstract

Given an accurate piecewise constant flux approximation $q$ of a smooth (but unknown) exact flux $p$, how can one compute a Sobolev norm of $p$ over a small domain $\omega$? This question arises in a duality argument of goal-oriented finite element a posteriori error analysis. Two equivalent practical estimators $\eta_{M}$ and $\eta_{\mathcal{E}}$ are presented for an approximation $\!q\!$ of $|p|_{H^1(\omega)}$ and model situations are addressed where $\!q\!$ is asymptotically an upper and +lower bound of $|p|_{H^1(\Omega)}$.