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
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 6, pp. 1023-1037
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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.
@article{ZVMMF_2010_50_6_a4,
author = {A. V. Razgulin},
title = {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},
journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
pages = {1023--1037},
publisher = {mathdoc},
volume = {50},
number = {6},
year = {2010},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZVMMF_2010_50_6_a4/}
}
TY - JOUR AU - A. V. Razgulin TI - 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 JO - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki PY - 2010 SP - 1023 EP - 1037 VL - 50 IS - 6 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/ZVMMF_2010_50_6_a4/ LA - ru ID - ZVMMF_2010_50_6_a4 ER -
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A. V. Razgulin. 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. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 6, pp. 1023-1037. http://geodesic.mathdoc.fr/item/ZVMMF_2010_50_6_a4/