A sparse algebraic multigrid method is studied as a cheap and accurate way to compute approximations of Schur complements of matrices arising from the discretization of some symmetric and positive definite partial differential operators. The construction of such a multigrid is discussed and numerical experiments are used to verify the properties of the method.
@article{M2AN_2003__37_1_133_0,
author = {Martikainen, Janne},
title = {Numerical study of two sparse AMG-methods},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {37},
year = {2003},
pages = {133-142},
doi = {10.1051/m2an:2003016},
mrnumber = {1972654},
zbl = {1030.65128},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2003__37_1_133_0}
}
Martikainen, Janne. Numerical study of two sparse AMG-methods. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 37 (2003) pp. 133-142. doi : 10.1051/m2an:2003016. http://gdmltest.u-ga.fr/item/M2AN_2003__37_1_133_0/
[1] , The finite element method with Lagrangian multipliers. Numer. Math. 20 (1972/73) 179-192. | Zbl 0258.65108
[2] , and, A new nonconforming approach to domain decomposition: the mortar element method, in Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. XI, Paris (1989-1991) 13-51. Longman Sci. Tech., Harlow (1994). | Zbl 0797.65094
[3] , and, The construction of preconditioners for elliptic problems by substructuring. I. Math. Comp. 47 (1986) 103-134. | Zbl 0615.65112
[4] , and Jinchao Xu, Parallel multilevel preconditioners. Math. Comp. 55 (1990) 1-22. | Zbl 0703.65076
[5] Qianshun Chang, Yau Shu Wong and Hanqing Fu, On the algebraic multigrid method. J. Comput. Phys. 125 (1996) 279-292. | Zbl 0857.65037
[6] , A capacitance matrix method for Dirichlet problem on polygon region. Numer. Math. 39 (1982) 51-64. | Zbl 0478.65062
[7] ,,, and, Distributed Lagrange multiplier methods for particulate flows, in Computational Science for the 21st Century, M.-O. Bristeau, G. Etgen, W. Fitzgibbon, J.L. Lions, J. Periaux and M.F. Wheeler Eds., Wiley (1997) 270-279. | Zbl 0919.76077
[8] , Tsorng-Whay Pan and J. Périaux, A fictitious domain method for Dirichlet problem and applications. Comput. Methods Appl. Mech. Engrg. 111 (1994) 283-303. | Zbl 0845.73078
[9] and, The use of preconditioning over irregular regions, in Computing methods in applied sciences and engineering VI, Versailles (1983) 3-14. North-Holland, Amsterdam (1984). | Zbl 0564.65067
[10] , Iterative methods for solving linear systems. SIAM, Philadelphia, PA (1997). | MR 1474725 | Zbl 0883.65022
[11] , Algebraic multi-grid for discrete elliptic second-order problems, in Multigrid methods V, Stuttgart (1996) 157-172. Springer, Berlin (1998). | Zbl 0926.65128
[12] , Efficient iterative solvers for elliptic finite element problems on nonmatching grids. Russian J. Numer. Anal. Math. Modelling 10 (1995) 187-211. | Zbl 0839.65031
[13] , Overlapping domain decomposition with non-matching grids. East-West J. Numer. Math. 6 (1998) 299-308. | Zbl 0918.65075
[14] , and, A moving mesh fictitious domain approach for shape optimization problems. ESAIM: M2AN 34 (2000) 31-45. | Numdam | Zbl 0948.65064
[15] , and, Multilevel preconditioners for Lagrange multipliers in domain imbedding. Electron. Trans. Numer. Anal. (to appear). | MR 1991267 | Zbl 1030.65129
[16] , A multilevel AINV preconditioner. Numer. Algorithms 29 (2002) 107-129. | Zbl 1044.65036
[17] and, Algebraic multigrid. SIAM, Philadelphia, PA, Multigrid methods (1987) 73-130.
[18] and, Fast iterative solution of stabilised Stokes systems. II. Using general block preconditioners. SIAM J. Numer. Anal. 31 (1994) 1352-1367. | Zbl 0810.76044
[19] ,, and, A domain decomposition preconditioner based on a change to a multilevel nodal basis. SIAM J. Sci. Statist. Comput. 12 (1991) 1486-1495. | Zbl 0744.65084