A weighted estimate for the rate of convergence of a projection-difference scheme for a parabolic equation and its application to the approximation of the initial-data control problem

A new technique is proposed for analyzing the convergence of a projection difference scheme as applied to the initial value problem for a linear parabolic operator-differential equation. The technique is based on discrete analogues of weighted estimates reflecting the smoothing property of solutions to the differential problem for $t>0$. Under certain conditions on the right-hand side, a new convergence rate estimate of order $O(\sqrt{\tau}+h)$ is obtained in a weighted energy norm without making any a priori assumptions on the additional smoothness of weak solutions. The technique leads to a natural projection difference approximation of the problem of controlling nonsmooth initial data. The convergence rate estimate obtained for the approximating control problems is of the same order $O(\sqrt{\tau}+h)$ as for the projection difference scheme.