Geodesic
Strong-Norm Error Estimates for the Projective-Difference Method for Parabolic Equations with Modified Crank–Nicolson Scheme
Matematičeskie zametki, Tome 74 (2003) no. 6, pp. 913-923 
V. V. Smagin. Strong-Norm Error Estimates for the Projective-Difference Method for Parabolic Equations with Modified Crank–Nicolson Scheme. Matematičeskie zametki, Tome 74 (2003) no. 6, pp. 913-923. http://geodesic.mathdoc.fr/item/MZM_2003_74_6_a11/
@article{MZM_2003_74_6_a11,
     author = {V. V. Smagin},
     title = {Strong-Norm {Error} {Estimates} for the {Projective-Difference} {Method} for {Parabolic} {Equations} with {Modified} {Crank{\textendash}Nicolson} {Scheme}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {913--923},
     year = {2003},
     volume = {74},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2003_74_6_a11/}
}
TY  - JOUR
AU  - V. V. Smagin
TI  - Strong-Norm Error Estimates for the Projective-Difference Method for Parabolic Equations with Modified Crank–Nicolson Scheme
JO  - Matematičeskie zametki
PY  - 2003
SP  - 913
EP  - 923
VL  - 74
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/MZM_2003_74_6_a11/
LA  - ru
ID  - MZM_2003_74_6_a11
ER  - 
%0 Journal Article
%A V. V. Smagin
%T Strong-Norm Error Estimates for the Projective-Difference Method for Parabolic Equations with Modified Crank–Nicolson Scheme
%J Matematičeskie zametki
%D 2003
%P 913-923
%V 74
%N 6
%U http://geodesic.mathdoc.fr/item/MZM_2003_74_6_a11/
%G ru
%F MZM_2003_74_6_a11

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

A parabolic problem in a separable Hilbert space is solved approximately by the projective-difference method. The problem is discretized with respect to space by the Galerkin method and with respect to time by the modified Cranck–Nicolson scheme. In this paper, we establish efficient (in time and space) strong-norm error estimates for approximate solutions. These estimates allow us to obtain the rate of convergence with respect to time of the error to zero up to the second order. In addition, the error estimates take into account the approximation properties of projective subspaces, which is illustrated for subspaces of finite element type.

[1] Smagin V. V., “Otsenki v silnykh normakh pogreshnosti proektsionno-raznostnogo metoda priblizhennogo resheniya abstraktnogo parabolicheskogo uravneniya”, Matem. zametki, 62:6 (1997), 898–909 | MR | Zbl

[2] Smagin V. V., “Srednekvadratichnye otsenki pogreshnosti proektsionno-raznostnogo metoda dlya parabolicheskikh uravnenii”, ZhVMiMF, 40:6 (2000), 908–919 | MR | Zbl

[3] Smagin V. V., “Energeticheskie otsenki pogreshnosti proektsionno-raznostnogo metoda so skhemoi Kranka–Nikolson dlya parabolicheskikh uravnenii”, Sib. matem. zh., 42:3 (2001), 670–682 | MR | Zbl

[4] Smagin V. V., “Proektsionno-raznostnye metody priblizhennogo resheniya parabolicheskikh uravnenii s nesimmetrichnymi operatorami”, Differents. uravneniya, 37:1 (2001), 115–123 | MR | Zbl

[5] Turetaev I. D., “Tochnye otsenki gradienta pogreshnosti proektsionno-raznostnykh skhem dlya parabolicheskikh uravnenii v proizvolnoi oblasti”, ZhVMiMF, 26:11 (1986), 1748–1751 | MR

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

[7] Smagin V. V., “O razreshimosti abstraktnogo parabolicheskogo uravneniya s operatorom, oblast opredeleniya kotorogo zavisit ot vremeni”, Differents. uravneniya, 32:5 (1996), 711–712 | MR | Zbl

[8] Smagin V. V., “O gladkoi razreshimosti variatsionnykh zadach parabolicheskogo tipa”, Tr. matem. f-ta (novaya seriya), no. 3, Izd-vo Voronezhskogo gos. un-ta, Voronezh, 1998, 67–72

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

[10] Vainikko G. M., Oya P. E., “O skhodimosti i bystrote skhodimosti metoda Galerkina dlya abstraktnykh evolyutsionnykh uravnenii”, Differents. uravneniya, 11:7 (1975), 1269–1277 | MR | Zbl

[11] Marchuk G. I., Metody vychislitelnoi matematiki, Nauka, M., 1989

[12] Smagin V. V., “Otsenki pogreshnosti poludiskretnykh priblizhenii po Galerkinu dlya parabolicheskikh uravnenii s kraevym usloviem tipa Neimana”, Izv. vuzov. Ser. matem., 1996, no. 3(406), 50–57 | MR | Zbl