Geodesic
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 
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/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Temam R., Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, Springer-Verlag, New York, 1997 | MR

[2] Lions Zh.-L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, Nauka, M., 1971 | Zbl

[3] Osipov Yu. S., Vasilev F. P., Potapov M. M., Osnovy metoda dinamicheskoi regulyarizatsii, Izd-vo MGU, M., 1999

[4] Babin A. V., Vishik M. I., Attractors of evolution equations, Studies in mathematics and its applications, 25, Elsevier, North Holland, 1992 | MR

[5] Zlotnik A. A., “Otsenka skorosti skhodimosti v $L_2$ proektsionno-raznostnykh skhem dlya parabolicheskikh uravnenii”, Zh. vychisl. matem. i matem. fiz., 18:6 (1978), 1454–1465 | MR | Zbl

[6] Zlotnik A. A., “Otsenka skorosti skhodimosti v $V_2(\mathcal{Q}_T)$ proektsionno-raznostnykh skhem dlya parabolicheskikh uravnenii”, Vestn. MGU. Ser. 15. Vychisl. matem. i kibernetika, 1980, no. 1, 27–36 | MR

[7] Smagin V. V., “Otsenka skorosti skhodimosti proektsionnogo i proektsionno-raznostnogo metodov dlya slabo razreshimykh parabolicheskikh uravnenii”, Matem. sb., 188:3 (1997), 143–160 | MR

[8] Smagin V. V., “Koertsitivnye otsenki pogreshnostei proektsionnogo i proektsionno-raznostnogo metodov dlya parabolicheskikh uravnenii”, Matem. sb., 185:11 (1994), 79–94 | Zbl

[9] Razgulin A. V., “Approksimatsiya zadachi upravleniya preobrazovaniem argumentov v nelineinom parabolicheskom uravnenii”, Zh. vychisl. matem. i matem. fiz., 41:12 (2001), 1844–1856 | MR | Zbl

[10] Razgulin A. V., “Proektsionno-raznostnaya skhema dlya parabolicheskogo funktsionalno-differentsialnogo uravneniya s dvumernym preobrazovaniem argumentov”, Zh. vychisl. matem. i matem. fiz., 45:10 (2005), 1848–1859 | MR | Zbl

[11] Syarle F., Metod konechnykh elementov dlya ellipticheskikh zadach, Mir, M., 1980 | MR

[12] Vasilev F. P., Metody resheniya ekstremalnykh zadach, Faktorial, M., 2001 | Zbl

[13] Pulin D. S., Razgulin A. V., “Distortion suppression optimization for a class of nonlinear optical systems with feedback”, Comput. Math. Modelling, 17:2 (2006), 155–171 | DOI | MR | Zbl