  Computable estimates for the error of finite element discretisations of
parabolic problems in the $L^\infty(0,T; L^2)$ norm are developed, which
exhibit constant effectivities (the ratio of the estimated error to the true
error) with respect to the simulation time. These estimates, which are of
optimal order, represent a significant advantage for long-time simulations, and
are derived using energy techniques based on elliptic reconstructions. The
effectivities of previous optimal order error estimates in this norm derived
using energy techniques are shown numerically to grow either in proportion to
the simulation duration or its square root, a key disadvantage compared with
earlier estimators derived using parabolic duality arguments. The new estimates
form a continuous family, almost all of which are new, reproducing certain
familiar energy-based estimates well suited for short-time simulations and not
available through the parabolic duality framework. For clarity, we demonstrate
the technique applied to a linear parabolic problem discretised using standard
conforming finite element methods in space coupled with backward Euler and
Crank-Nicolson time discretisations, although it can be applied much more
widely.
