, Volume 56, Issue 2, pp 157177
First online:
Algebraic multilevel preconditioning methods. I
 O. AxclssonAffiliated withDepartment of Mathematics, Catholic University
 , P. S. VassilevskiAffiliated withDepartment of Mathematics, Catholic University
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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.
Subject Classifications
AMS(MOS): 65F10 65N20 65N30 CR: G1.3 Title
 Algebraic multilevel preconditioning methods. I
 Journal

Numerische Mathematik
Volume 56, Issue 23 , pp 157177
 Cover Date
 198902
 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