A posteriori error estimates for the fractional-step ϑ-scheme for linear parabolic equations

We derive residual-based a posteriori error estimates of optimal order for time discretizations of linear parabolic equations by the fractional-step ϑ-scheme. We first consider the time semidiscrete problem. The main tool of our analysis is an appropriate reconstruction of the piecewise linear interpolant of the approximate solution that leads to a residual of optimal order. Next we extend the above-mentioned results to the case of a full discretization. The theoretical results are justified with numerical experiments.