Weighted estimate for the convergence rate 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(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqrt \tau $$\end{document} + 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(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqrt \tau $$\end{document} + h) as for the projection difference scheme.