A posteriori error estimates for fully discrete fractional-step $\vartheta$-approximations for the heat equation

We derive optimal order a posteriori error estimates for fully discrete approximations of the initial-boundary value problem for the heat equation. For the discretization in time we apply the fractional-step $\vartheta$-scheme and for the discretization in space the finite element method with finite element spaces that are allowed to change with time. The first optimal order a posteriori error estimates in $L^\infty(0, T ; L^2({\varOmega}))$ are derived by applying the reconstruction technique.