Mathematical Models and Computer Simulations

, Volume 11, Issue 3, pp 426–437

• V. T. Zhukov
• N. D. Novikova
• O. B. Feodoritova
Article

Abstract

An adaptive Chebyshev iterative method used to solve boundary-value problems for three-dimensional elliptic equations numerically is constructed. In this adaptive method, the unknown lower bound of the spectrum of the discrete operator is refined in the additional iteration cycle, and the upper bound of the spectrum is taken to be its estimate by the Gershgorin theorem. Such a procedure ensures the convergence of the constructed adaptive method with the computational costs close to the costs of the standard Chebyshev method, which uses the exact bounds of the spectrum of the discrete operator.

Notes

ACKNOWLEDGMENTS

This study was supported by a grant of the Russian Science Foundation (project no. 14-21-00025-P).

REFERENCES

1. 1.
V. T. Zhukov, N. D. Novikova, and O. B. Feodoritova, “Multigrid method for anisotropic diffusion equations based on adaptive Chebyshev smoothers,” Math. Models Comput. Simul. 7, 117–127 (2015).
2. 2.
V. T. Zhukov, N. D. Novikova, and O. B. Feodoritova, “Multigrid method for elliptic equations with anisotropic discontinuous coefficients,” Comput. Math. Math. Phys. 55, 1150–1163 (2015).
3. 3.
V. T. Zhukov, N. D. Novikova, and O. B. Feodoritova, “On the solution of evolution equations based on multigrid and explicit iterative methods,” Comput. Math. Math. Phys. 55, 1276–1289 (2015).
4. 4.
V. T. Zhukov, M. M. Krasnov, N. D. Novikova, and O. B. Feodoritova, “Algebraic multigrid method with adaptive smoothers based on Chebyshev polynomials,” KIAM Preprint No. 113 (Keldysh Inst. Appl. Math., Moscow, 2016). http://library.keldysh.ru/preprint.aspıd=2016-113.
5. 5.
A. H. Baker, R. D. Falgout, T. Gamblin, T. V. Kolev, M. Schulz, and U. M. Yang, “Scaling algebraic multigrid solvers: on the road to exascale,” in Competence in High Performance Computing (Springer, Berlin, Heidelberg, 2010), pp. 215–226.Google Scholar
6. 6.
A. Baker, R. Falgout, T. Kolev, and U. Yang, “Multigrid smoothers for ultra-parallel computing,” SIAM J. Sci. Comput. 33, 2864–2887 (2011).
7. 7.
F. R. Gantmacher, The Theory of Matrices (AMS, Chelsea, 2000; Nauka, Moscow, 1966).Google Scholar
8. 8.
A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations, Vol. 1: Direct Methods, Vol. 2: Iterative Methods (Birkhauser, Basel, Boston, Berlin, 1989; Nauka, Moscow, 1978).Google Scholar
9. 9.
P. L. Chebyshev, “Questions about the smallest values associated with the approximate representation of functions,” in Writings (Sand Pb., 1899), Vol. 1, pp. 705–710 [in Russian].Google Scholar
10. 10.
A. S. Shvedov and V. T. Zhukov, “Explicit iterative difference schemes for parabolic equations,” Russ. J. Numer. Anal. Math. Model. 13, 133–148 (1998).
11. 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,” R. Soc. Philos. Trans. A 210, 307–357 (1910).
12. 12.
M. K. Gavurin, “Application of best approximation polynomials to improving convergence of iterative processes,” Usp. Mat. Nauk 5, 156–160 (1950).Google Scholar
13. 13.
D. Flanders and G. Shortley, “Numerical determination of fundamental modes,” Appl. Phys. 21, 1326–1332 (1950).
14. 14.
G. H. Shortley, “Use of Tschebyscheff-polynomial operators in the solution of boundary value problems,” J. Appl. Phys. 24, 392-396 (1953).
15. 15.
D. M. Young, “On Richardson’s method for solving linear systems with positive definite matrices,” Math. Phys. 32, 243–255 (1954).
16. 16.
V. I. Lebedev and S. A. Finogenov, “On the order of selection of iterative parameters in the Chebyshev cyclic method,” Zh. Vychisl. Mat. Mat. Fiz. 11, 425–438 (1971).
17. 17.
E. S. Nikolaev and À. À. Samarskii, “Selection of iterative parameters in the Richardson method,” Zh. Vychisl. Mat. Mat. Fiz. 12, 960–973 (1972).
18. 18.
Yuan Čžao-din, “Some difference schemes for solving the first boundary value problem for linear partial differential equations,” Cand. Sci. (Phys. Math.) Dissertation (Mosc. State Univ., Moscow, 1958).Google Scholar
19. 19.
Yuan Čžao-din, “Some difference schemes for the numerical solution of a parabolic differential equation,” Mat. Sb. 50, 391–422 (1960).
20. 20.
Yuan Čžao-din, “On the stability of difference schemes for solving differential equations of parabolic type,” Dokl. Akad. Nauk SSSR 117, 578–581 (1957).
21. 21.
V. K. Saul’ev, Integration of Parabolic Equations by the Grid Method, Ed. by L. A. Lyusternik (Fizmatgiz, Moscow, 1960) [in Russian].Google Scholar
22. 22.
L. A. Lyusternik, “Remarks on the numerical solution of boundary problems of the Laplace equation and the calculation of eigenvalues by the grid method,” Tr. Mat. Inst. Steklova 20, 49-64 (1947).
23. 23.
N. S. Bakhvalov, N. P. Zhidkov, and G. M. Kobelkov, Numerical Methods (Nauka, Moscow, 1987) [in Russian].Google Scholar
24. 24.
J. Demmel, Applied Numerical Linear Algebra (SIAM, Philadelphia, 1997).
25. 25.
R. Eymard, G. Henry, R. Herbin, F. Hubert, R. Klofkorn, and G. Manzini, “3D benchmark on discretization schemes for anisotropic diffusion problems on general grids,” HAL Id: hal-00580549 (2011). https://hal.archives-ouvertes.fr/hal-0058054.Google Scholar
26. 26.
V. T. Zhukov, N. D. Novikova, and O. B. Feodoritova, An Adative Cebysev Method (Ross. Akad. Nauk, Moscow, 2017) [in Russian].Google Scholar