  In this paper we consider a sub-diffusion problem where the fractional time
derivative is approximated either by the L1 scheme or by Convolution
Quadrature. We propose new interpretations of the numerical schemes which lead
to a posteriori error estimates. Our approach is based on appropriate pointwise
representations of the numerical schemes as perturbed evolution equations and
on stability estimates for the evolution equation. A posteriori error estimates
in $L^2(H)$ and $L^\infty (H)$ norms of optimal order are derived. Extensive
numerical experiments indicate the reliability and the optimality of the
estimators for the schemes considered, as well as their efficiency as error
indicators driving adaptive mesh selection locating singularities of the
problem.
