Algebraic multilevel preconditioning methods. I
 O. Axclsson,
 P. S. Vassilevski
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessSummary
A recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed. It is based on a sequence of nested finite element spaces with the usual nodal basis functions. Using a nodeordering corresponding to the nested meshes, the finite element stiffness matrix is recursively split up into twolevel block structures and is factored approximately in such a way that any successive Schur complement is replaced (approximated) by a matrix defined recursively and thereform only implicitely given. To solve a system with this matrix we need to perform a fixed number (v) of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level. It is shown that by a proper choice of iteration parameters it suffices to use \(v > \left( {1  \gamma ^2 } \right)^{  \tfrac{1}{2}} \) iterations for the so constructedvfoldVcycle (wherev=2 corresponds to aWcycle) preconditioning matrices to be spectrally equivalent to the stiffness matrix. The conditions involve only the constant λ in the strengthened C.B.S. inequality for the corresponding twolevel hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance. If we use successive uniform refinements of the meshes the method is of optimal order of computational complexity, if \(\gamma ^2< \tfrac{8}{9}\) . Under reasonable assumptions of the finite element mesh, the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method, for instance.
 Axelsson, O., Barker, V.A.: Finite element solution of boundary value problems. 1st Ed. New York: Academic Press 1984
 Axelsson, O., Gustafsson, I.: Preconditioning and twolevel multigrid methods of arbitrary degree of approximation. Math. Comput.40, 219–242 (1983)
 Axelsson, O.: On multigrid methods of the twolevel type. In: Hackbusch, W., Trottenberg, U. (eds.) Multigrid methods, Proceedings, KölnPorz 1981, pp. 352–367. LNM 960, Berlin Heidelberg New York: Springer 1982
 Bank, R., Dupont, T.: Analysis of a twolevel scheme for solving finite element equations, Report CNA159, Center for Numerical Analysis, The University of Texas at Austin 1980
 Bank, R., Dupont, T.: An optimal order process for solving finite element equations. Math. Comput.36, 35–51 (1981)
 Bank, R., Dupont, T., Yserentant, H.: The hierarchical basis multigrid method. Numer. Math.52, 427–458 (1988)
 Braess, D.: The contraction number of a multigrid method for solving the Poisson equation. Numer. Math.37, 387–404 (1981)
 Braess, D.: The convergence rate of a multigrid method with GaussSeidel relaxation for the Poisson equation. In: Hackbusch, W., Trottenberg, U. (eds.) Multigrid methods, Proceedings, KölnPorz 1981, pp. 368–386. LNM 960, Berlin Heidelberg New York: Springer 1982
 Braess, D., Hackbusch, W.: A new convergence proff for the multigrid method including theVcycle. SIAM J. Numer. Anal20, 967–975 (1983)
 George, A., Liu, J.W.: Computer solution of large sparse positive definite systems, 1st Ed. Englewood Cliffs NJ: PrenticeHall 1984
 Kuznetsov, Y.A.: Multigrid domain decomposition methods for elliptic problems, talk presented at the conference on Numerical Linear Algebra and Parallel Computation, Mathematisches Forschungsinstitut, Oberwolfach, Febr. 28March 5 (1988)
 Maitre, J.F., Musy, F.: The contraction number of a class of twolevel methods; an exact evaluation for some finite element subspaces and model problems. In: Hackbusch, W., Trottenberg, U. (eds.) Multigrid methods, Proceedings, KölnPorz 1981, pp. 535–544. LNM 960, Springer, 1982
 Vassilevski, P.: Nearly optimal iterative methods for solving finite element equations based on multilevel splitting of the matrix 1987 (submitted)
 Yserentant, H.: On the multilevel splitting of finite element spaces. Numer. Math.49, 379–412 (1986)
 Title
 Algebraic multilevel preconditioning methods. I
 Journal

Numerische Mathematik
Volume 56, Issue 23 , pp 157177
 Cover Date
 19890201
 DOI
 10.1007/BF01409783
 Print ISSN
 0029599X
 Online ISSN
 09453245
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 AMS(MOS): 65F10
 65N20
 65N30
 CR: G1.3
 Industry Sectors
 Authors

 O. Axclsson ^{(1)}
 P. S. Vassilevski ^{(1)}
 Author Affiliations

 1. Department of Mathematics, Catholic University, Toernooiveld, 6525 ED, Nijmegen, The Netherlands