On development of parallel algorithms for the solution of parabolic and elliptic equations
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Mathematical Analysis, Tome 155 (2018), pp. 20-37.

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

Results of the development of certain parallel numerical methods of solution of three-dimensional evolutionary and stationary problems of diffusion and thermal conductivity are given in the paper. We present a detailed description of a special explicit iteration scheme for parabolic equations and discuss a multigrid technology used for the solution of elliptic equations and implicit schemes for parabolic equations.
Mots-clés : parabolic equations, elliptic equations, adaptation.
Keywords: multigrid method, Chebyshev parameters
@article{INTO_2018_155_a1,
     author = {V. T. Zhukov and O. B. Feodoritova},
     title = {On development of parallel algorithms for the solution of parabolic and elliptic equations},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {20--37},
     publisher = {mathdoc},
     volume = {155},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2018_155_a1/}
}
TY  - JOUR
AU  - V. T. Zhukov
AU  - O. B. Feodoritova
TI  - On development of parallel algorithms for the solution of parabolic and elliptic equations
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2018
SP  - 20
EP  - 37
VL  - 155
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2018_155_a1/
LA  - ru
ID  - INTO_2018_155_a1
ER  - 
%0 Journal Article
%A V. T. Zhukov
%A O. B. Feodoritova
%T On development of parallel algorithms for the solution of parabolic and elliptic equations
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2018
%P 20-37
%V 155
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2018_155_a1/
%G ru
%F INTO_2018_155_a1
V. T. Zhukov; O. B. Feodoritova. On development of parallel algorithms for the solution of parabolic and elliptic equations. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Mathematical Analysis, Tome 155 (2018), pp. 20-37. http://geodesic.mathdoc.fr/item/INTO_2018_155_a1/

[1] Bakhvalov N. S., Zhidkov N. P., Kobelkov G. M., Chislennye metody, Nauka, M., 1987 | MR

[2] Gantmakher F. R., Teoriya matrits, Nauka, M., 1966 | MR

[3] Gelfand I. M., Lokutsievskii O. V., “O raznostnykh skhemakh dlya resheniya uravneniya teploprovodnosti”, Vvedenie v teoriyu raznostnykh skhem, eds. Godunov S. K., Ryabenkii B. C., Fizmatgiz, M., 1962

[4] Demmel Dzh., Vychislitelnaya lineinaya algebra. Teoriya i prilozheniya, Mir, M., 2001

[5] Zhukov V. T., Novikova N. D., Feodoritova O. B., “Parallelnyi mnogosetochnyi metod dlya raznostnykh ellipticheskikh uravnenii”, Mat. model., 26:1 (2014), 55–68 | Zbl

[6] Zhukov V. T., Novikova N. D., Feodoritova O. B., “Mnogosetochnyi metod dlya anizotropnykh uravnenii diffuzii na osnove adaptatsii chebyshevskikh sglazhivatelei”, Mat. model., 26:9 (2014), 126–140 | Zbl

[7] Zhukov V. T., Novikova N. D., Feodoritova O. B., “Mnogosetochnyi metod dlya ellipticheskikh uravnenii s anizotropnymi razryvnymi koeffitsientami”, Zh. vychisl. mat. mat. fiz., 55:7 (2015), 1168 | DOI | Zbl

[8] Zhukov V. T., Novikova N. D., Feodoritova O. B., “O reshenii evolyutsionnykh uravnenii mnogosetochnym i yavno-iteratsionnym metodami”, Zh. vychisl. mat. mat. fiz., 55:8 (2015), 1305–1319 | DOI | Zbl

[9] Zhukov V. T. Krasnov M. M., Novikova N.D., Feodoritova O. B., “Sravnenie effektivnosti mnogosetochnogo metoda na sovremennykh vychislitelnykh arkhitekturakh”, Programmirovanie, 2015, no. 1, 21–31

[10] Zhukov V. T., “O yavnykh metodakh chislennogo integrirovaniya dlya parabolicheskikh uravnenii”, Mat. model., 22:10 (2010), 127–158 | Zbl

[11] Zhukov V. T. Krasnov M. M., Novikova N. D., Feodoritova O. B., Algebraicheskii mnogosetochnyi metod c adaptivnymi sglazhivatelyami na osnove mnogochlenov Chebysheva, IPM im. M. V. Keldysha, M., 2016

[12] Lebedev V. I., Finogenov S. A., “O poryadke vybora iteratsionnykh parametrov v chebyshevskom tsiklicheskom metode”, Zh. vychisl. mat. mat. fiz., 11:2 (1971), 425–438 | MR | Zbl

[13] Lokutsievskii V. O., Lokutsievskii O. V., “O chislennom reshenii kraevykh zadach dlya uravnenii parabolicheskogo tipa”, Dokl. AN SSSR, 291:3 (1986), 540–544 | MR

[14] Lyusternik L. A., “Zamechaniya k chislennomu resheniyu kraevykh zadach uravneniya Laplasa i vychisleniyu sobstvennykh znachenii metodom setok”, Tr. Mat. in-ta im. V. A. Steklova, 20 (1947), 49–64 | MR | Zbl

[15] Samarskii A. A., Teoriya raznostnykh skhem, Nauka, M., 1977 | MR

[16] Samarskii A. A., Nikolaev E. S., Metody resheniya setochnykh uravnenii, Nauka, M., 1978

[17] Fedorenko R. P., “Relaksatsionnyi metod resheniya raznostnykh ellipticheskikh uravnenii”, Zh. vychisl. mat. mat. fiz., 1:5 (1961), 922–927 | Zbl

[18] Khorn R., Dzhonson Ch., Matrichnyi analiz, Mir, M., 1989 | MR

[19] Chebyshev P. L., “Voprosy o naimenshikh velichinakh, svyazannye s priblizhennym predstavleniem funktsii”, Polnoe sobranie sochinenii, v. 2, ed. Chebyshev P. L., M., L., 1947, 151–235 | MR

[20] Ruge J. W., Stuben K., “Algebraic Multigrid (AMG)”, Multigrid Methods, Front. Appl. Math., 3, ed. S. F. McCormick, SIAM, 1987, 73–130 | MR

[21] Trottenberg U., Oosterlee C. W., Schuller A., Multigrid, Academic Press, 2001 | MR | Zbl