Resumen
It is known that the energy technique for a posteriori error analysis of finite element discretizations of parabolic problems yields suboptimal rates in the norm $L^\infty (0,T; L^2 (\Omega)).$ In this paper, we combine energy techniques with an appropriate pointwise representation of the error based on an elliptic reconstruction operator which restores the optimal order (and regularity for piecewise polynomials of degree higher than one). This technique may be regarded as the "dual a posteriori" counterpart of Wheeler's elliptic projection method in the a priori error analysis.

Resumen
We consider semidiscrete and fully discrete conservative finite volume schemes approximating the solution to one-dimensional scalar conservation law. We show that all E-schemes are associated with a discrete kinetic formulation with a nonnegative kinetic defect measure. This construction provides an alternative proof of the discrete local entropy inequalities with simple expressions of the discrete entropy fluxes. In contrast to the known results, which are restricted to CFL of the form $\lambda Q\leq 1/2$, our proof holds under "sharp" CFL conditions.