Geodesic
A multigrid method for the heat equation with discontinuous coefficients with the special choice of grids
Matematičeskoe modelirovanie, Tome 27 (2015) no. 9, pp. 17-32.

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

For the solution of systems of linear algebraic equations obtained as a result of discretization of initial-boundary value problems for the heat equation with discontinuous heat conduction coefficient, a new mutigrid method is proposed. In the method, a special construction of the next level grid is used, with special treatment of sub-regions near the discontinuity lines of the heat conduction coefficient. Numerical experiments with 2D model problem discretized on orthogonal grids demonstrated a high speed of convergence for the method and weak dependence of the convergence on the discontinuity jump of the coefficient.
Mots-clés : parabolic equations
Keywords: multigrid methods, speed of convergence of an iterative method. 
@article{MM_2015_27_9_a1,
     author = {O. Yu. Milyukova and V. F. Tishkin},
     title = {A multigrid method for the heat equation with discontinuous coefficients with the special choice of grids},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {17--32},
     publisher = {mathdoc},
     volume = {27},
     number = {9},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2015_27_9_a1/}
}
TY  - JOUR
AU  - O. Yu. Milyukova
AU  - V. F. Tishkin
TI  - A multigrid method for the heat equation with discontinuous coefficients with the special choice of grids
JO  - Matematičeskoe modelirovanie
PY  - 2015
SP  - 17
EP  - 32
VL  - 27
IS  - 9
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2015_27_9_a1/
LA  - ru
ID  - MM_2015_27_9_a1
ER  - 
%0 Journal Article
%A O. Yu. Milyukova
%A V. F. Tishkin
%T A multigrid method for the heat equation with discontinuous coefficients with the special choice of grids
%J Matematičeskoe modelirovanie
%D 2015
%P 17-32
%V 27
%N 9
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2015_27_9_a1/
%G ru
%F MM_2015_27_9_a1
O. Yu. Milyukova; V. F. Tishkin. A multigrid method for the heat equation with discontinuous coefficients with the special choice of grids. Matematičeskoe modelirovanie, Tome 27 (2015) no. 9, pp. 17-32. http://geodesic.mathdoc.fr/item/MM_2015_27_9_a1/

[1] A. A. Samarskii, The Theory of Difference Schemes, Marcel Dekker, New York, 2001 | MR | MR | Zbl

[2] N. S. Bakhvalov, N. P. Zhidkov, G. M. Kobelkov, Chislennye metody, Nauka, M., 1987, 599 pp. | MR

[3] R. P. Fedorenko, “Relaksatsyonnyi metod resheniia raznostnykh ellipticheskikh uravnenii”, Zh. Vychisl. Mat. i Mat. Fiz., 1:5 (1961), 922–927 | MR | Zbl

[4] R. P. Fedorenko, “Skorost skhodimosti odnogo iteratsyonnogo metoda”, Zh. Vychisl. Mat. i Mat. Fiz., 4:3 (1964), 227–235

[5] N. S. Bakhvalov, “O skhodimosti odnogo relaksatsyonnogo metoda dlia ellipticheskjgo operatora s ectestvennymi ogranicheniiami”, Zh. Vychisl. Mat. i Mat. Fiz., 6:5 (1966), 101–135 | Zbl

[6] G. P. Astrahantsev, “Ob odnom relaksatsionnom metode”, Zh. Vychisl. Mat. i Mat. Fiz., 11:2 (1971), 439–448 | MR | Zbl

[7] A. Brandt, “Multi-level adaptive solutions to boundary value problems”, Math. Comp., 31 (1977), 333–390 | MR | Zbl

[8] W. Hackbusch, Multi-grid Methods and Applications, Springer, Berlin–Heidelberg, 1985, 773 pp. | Zbl

[9] V. V. Shaidurov, Mnogosetochnye metody konechnykh elementov, Nauka, M., 1989, 288 pp. | MR

[10] Yu. V. Vasilevskii, M. A. Olshanskii, Kratkii kurs po mnogosetochnym metodam i metodam dekompozitsii oblasti, MGU im. M. V. Lomonosova, fak. VMiK, M., 2007, 102 pp.

[11] A. Brandt, S. F. McCormick, J. W. Ruge, “Algebraic multigrid (AMG) for sparce matrix equations”, Sparsity and Its Applications, ed. D. J. Evans, Cambridge University Press, Cambridge, UK, 1985, 257–284 | MR

[12] K. Stüben, “An introduction to algebraic multigrid”, Multigrid, Academic press, San Diego, CA, 2001, 413–532

[13] Y. Notey, “An aggregation-based algebraic multigrid method”, Electronic Transaction on Numerical Analysis, 37 (2010), 123–146 | MR

[14] J. H. Bramble, J. E. Pasciak, J. Xu, “The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms”, Math. Comp., 56 (1991), 1–34 | MR | Zbl

[15] M. E. Ladonkina, O. Yu. Milyukova, V. F. Tishkin, “Numerical Algorithm for Solving Diffusion-Type Equations on the Basis of Multigrid Metods”, Comp. Math. and Math. Ph., 50:8 (2010), 1367–1390 | MR | Zbl

[16] O. Milyukova, M. Ladonkina, V. Tishkin, “A numerical method for solving parabolic equations based on the use of a multigrid method”, International Journal of Numerical Analysis Modeling, Series B, 2:2–3 (2012), 183–199 | MR

[17] O. Yu. Milyukova, V. F. Tishkin, “Chislennyi metod resheniia uravneniia teploprovodnosti na treugolnykh setkakh na osnove mnogosetochnogo metoda”, Keldysh Institute preprints, 2011, 029, 16 pp.

[18] O. Yu. Milyukova, V. F. Tishkin, “Chislennyi metod resheniia uravneniia teploprovodnosti s razryvnym koeffitsientom na osnove mnogosetochnogo metoda”, Keldysh Institute preprints, 2013, 064, 19 pp. | Zbl

[19] Y. Saad, Iterative methods for sparse linear systems, PWS Publishing Co., Int. Tompson Publ. Co., 1995, 447 pp.