All
 
|
 
 


An adaptive chebyshev iterative method
 
PIIS023408790001921-7-1
DOI10.31857/S023408790001921-7
Publication type Article
Status Published
Authors
V. Zhukov 
Affiliation: Keldysh Institute of Applied Mathematics of RAS
Address: Russian Federation, Moscow
Olga Feodoritova
Affiliation: Keldysh Institute of Applied Mathematics of RAS
Address: Russian Federation, Moscow
Natalia Novikova
Affiliation: Keldysh Institute of Applied Mathematics of RAS
Address: Russian Federation, Moscow
Journal nameMatematicheskoe modelirovanie
EditionVolume 30 Number 10
Pages67-85
Abstract

For the numerical solution of a boundary-value problem of three-dimensional elliptic equations an adaptive Chebyshev iterative method is constructed. In this adaptive method, the unknown lower bound of the spectrum of the discrete operator is refined in the additional cycle of the iterative method; the upper bound of the spectrum is taken to be its estimate by the Gershgorin theorem. Such procedure ensures the convergence of the constructed adaptive method with computational costs close to the costs of the Chebyshev method, which uses the exact boundaries of the spectrum of the discrete operator.

Keywordselliptic equations, Chebyshev polynomials, adaptive method
AcknowledgmentThe work was carried out at the expense of a grant from the RNF (project No. 14–21–00025 – П)
Received08.11.2018
Publication date14.11.2018
Cite   Download pdf To download PDF you should sign in
Размещенный ниже текст является ознакомительной версией и может не соответствовать печатной

views: 1552

Readers community rating: votes 0 

1. V.T. Zhukov, N.D. Novikova, O.B. Feodoritova. Mnogosetochnyj metod dlya anizotropnykh uravnenij diffuzii na osnove adaptatsii chebyshevskikh sglazhivatelej // Matem. modelirovanie, 2014, t.26, № 9, s.126?140; 
2. V.T. Zhukov, N.D. Novikova, O.B. Feodoritova. Mnogosetochnyj metod dlya ehllipti-cheskikh uravnenij s anizotropnymi razryvnymi koehffitsientami // Zh. vychisl. matem. i matem. fiz., 2015, t.55, № 7, s.1168–1182; 
3. V.T. Zhukov, N.D. Novikova, O.B. Feodoritova. O reshenii ehvolyutsionnykh uravnenij mnogosetochnym i yavno-iteratsionnym metodami // Zh. vychisl. matem. i matem. fiz., 2015, t.55, №8, s.1305–1319; 
4. V.T. Zhukov, M.M. Krasnov, N.D. Novikova, O.B. Feodoritova. Algebraicheskij mnogosetochnyj metod c adaptivnymi sglazhivatelyami na osnove mnogochlenov Chebysheva // Preprinty IPM im. M.V.Keldysha, 2016, № 113, 32 s. doi:10.20948/ prepr-2016-113. URL: http://library.keldysh.ru/preprint.asp?id=2016–113;  
5. A.H. Baker, R.D. Falgout, T. Gamblin, T.V. Kolev, M. Schulz, U.M. Yang. (2011) Scaling Algebraic Multigrid Solvers: On the Road to Exascale. In:Competence in High Performance Computing, 2010. Springer, Berlin, Heidelberg, p. 215-226.
6. A. Baker, R. Falgout, T. Kolev, U. Yang. Multigrid smoothers for ultra-parallel computing. SIAM J. Sci. Comput., 2011, v.33, №5, p.2864–2887.
7. F.R. Gantmakher. Teoriya matrits. – M.: Nauka, 1966, 576 s.; F.R. Gantmacher. The Theory of Matrices // AMS Chelsea Publishing: Reprinted by American Mathematical Society, 2000, 660 s. ISBN 0821813765.
8. A.A. Samarskij, E.S. Nikolaev. Metody resheniya setochnykh uravnenij. – M.: Nauka, 1978, 592 s.;.
9. P.L. Chebyshev. Voprosy o naimen'shikh velichinakh, svyazannye s priblizhennym predstavleniem funktsij. – SPb.: Sochineniya, 1899, t.1, 1899, s.705–710; 
10. A.S. Shvedov, V.T. Zhukov. Explicit iterative difference schemes for parabolic equations // Russian J. Numer. Anal. Math. Modeling, 1998, v.13, № 2, s.133–148.
11. L.F. Richardson. The approximate arithmetical solution by finite differences of physical problems involving differential equations with an application to the stresses in a masonry dam // Roy. Soc. Philos. Trans., 1910, 210A, p.307-357.
12. M.K. Gavurin. Primenenie polinomov nailuchshego priblizheniya k uluchsheniyu skhodimosti iterativnykh protsessov // UMN, 1950, t.5, № 3(37), s.156–160; 
13. D. Flanders and G. Shortley. Numerical determination of fundamental modes // Appl. Phys., 1950, v.21, № 12, p.1326–1332.
14. G.H. Shortley. Use of Tschebyscheff–polynomial operators in the solution of boundary value problems // J. Appl. Phys., 1953, v.24, p.392–396.
15. D.M. Young. On Richardson's method for solving linear systems with positive definitematrices // Math. Phys., 1954, 32, № 4, p.243–255.
16. V.I. Lebedev, S.A. Finogenov. O poryadke vybora iteratsionnykh parametrov v chebyshevskom tsiklicheskom metode // ZhVMiMF, 1971, t.11, № 2, s.425–438; 
17. E.S. Nikolaev, A.A. Samarskij. Vybor iteratsionnykh parametrov v metode Richardsona // Zh. vychisl. matem. i matem. fiz., 1972, t.12, № 4, s.960–973; 
18. Yuan' Chzhao-din. Nekotorye raznostnye skhemy resheniya pervoj kraevoj zadachi dlya linejnykh differentsial'nykh uravnenij s chastnymi proizvodnymi // Diss. na soisk. uchenoj stepeni kand. fiz.-mat. nauk. – M.: MGU, 1958; 
19. Yuan' Chzhao-din. Nekotorye raznostnye skhemy chislennogo resheniya differentsial'nogo uravneniya parabolicheskogo tipa // Matem. sb., 1960, t.50 (92), №4, s.391–422; 
20. Yuan' Chzhao-din. Ob ustojchivosti raznostnykh skhem dlya resheniya differentsial'nykh uravnenij parabolicheskogo tipa // DAN SSSR, 1957, t.117, № 4, c.578?581; 
21. V.K. Saul'ev. Integrirovanie uravnenij parabolicheskogo tipa metodom setok / Red. Lyusternik L. A. – M.: Fizmatgiz, 1960, 324 s.; 
22. L.A. Lyusternik. Zamechaniya k chislennomu resheniyu kraevykh zadach uravneniya Laplasa i vychisleniyu sobstvennykh znachenij metodom setok // Tr. Matem. in-ta im. V.A. Steklova, 1947, t.20, s.49–64; 
23. N.S. Bakhvalov, N.P. Zhidkov, G.M. Kobel'kov. Chislennye metody. – M.: Nauka, 1987, 600s.
24. Dzh. Demmel'. Vychislitel'naya linejnaya algebra. Teoriya i prilozheniya. – M.: Mir, 2001,430 s.
25. R. Eymard, G. Henry, R. Herbin, F. Hubert, R. Klofkorn, G. Manzini. 3D Benchmark on Discretization Schemes for Anisotropic Diffusion Problems on General Grids. HAL Id: hal–00580549 https://hal.archives-ouvertes.fr/hal-0058054.  2011.
26. V.T. Zhukov, N.D. Novikova, O.B. Feodoritova. Adaptivnyj chebyshevskij algoritm // Preprint IPM im. M.V. Keldysha RAN, 2017, 32 s.

Система Orphus

Loading...
Up