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

Abstract

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.