A comparison of duality and energy a posteriori estimates for $\mathrm {L}_{\infty }(0,T;\mathrm {L}_2(\varOmega ))$ in parabolic problems

We use the elliptic reconstruction technique in combination with a duality approach to prove a posteriori error estimates for fully discrete backward Euler scheme for linear parabolic equations. As an application, we combine our result with the residual based estimators from the a posteriori estimation for elliptic problems to derive space-error indicators and thus a fully practical version of the estimators bounding the error in the $ \mathrm {L}_{\infty }(0,T;\mathrm {L}_2(\varOmega ))$ norm. These estimators, which are of optimal order, extend those introduced by Eriksson and Johnson in 1991 by taking into account the error induced by the mesh changes and allowing for a more flexible use of the elliptic estimators. For comparison with previous results we derive also an energy-based a posteriori estimate for the $ \mathrm {L}_{\infty }(0,T;\mathrm {L}_2(\varOmega ))$-error which simplifies a previous one given by Lakkis and Makridakis in 2006. We then compare both estimators (duality vs. energy) in practical situations and draw conclusions.

Meat quality and health implications of organic and conventional beef production

Recommendation to reduce fat consumption from ruminant meat does not consider the contribution of nutritionally beneficial fatty acids in lean beef. Here we report effects of production system (organic vs conventional) and finishing season on meat and fat quality of sirloin steaks from retail outlets and simulated fatty acid intakes by consumers. There was little difference in meat quality (pH, shear force and colour), but the fat profiles varied considerably between production systems and season. Meat fat from organic and summer finished cattle contained higher concentrations of conjugated linoleic acid, its precursor vaccenic acid and individual omega-3 fatty acids and had a lower ratio of omega-6 to omega-3 fatty acids compared with non-organic and winter finished cattle respectively. The fat profile from summer finished organic beef aligns better to recommended dietary guideline including those for long chain omega-3 fatty acids compared with that from winter finished, non-organic steak.