Proceedings of the 1st International Conference on Supercomputing
Algebraic multilevel preconditioning methods, II
SIAM Journal on Numerical Analysis
Iterative solution methods
SIAM Journal on Scientific Computing
Multilevel preconditioning of 2D Rannacher-Turek FE problems: additive and multiplicative methods
NMA'06 Proceedings of the 6th international conference on Numerical methods and applications
Algebraic multilevel methods with aggregations: an overview
LSSC'05 Proceedings of the 5th international conference on Large-Scale Scientific Computing
New robust nonconforming finite elements of higher order
Applied Numerical Mathematics
Mathematics and Computers in Simulation
| Hi-index | 0.09 |
Preconditioners based on various multilevel extensions of two-level finite element methods (FEM) lead to iterative methods which often have an optimal order computational complexity with respect to the number of degrees of freedom of the system. Such methods were first presented in [O. Axelsson, P.S. Vassilevski, Algebraic multilevel preconditioning methods I, Numer. Math. 56 (1989) 157-177; O. Axelsson, P.S. Vassilevski, Algebraic multilevel preconditioning methods II, SIAM J. Numer. Anal. 27 (1990) 1569-1590] on (recursive) two-level splittings of the finite element space. The key role in the derivation of optimal convergence rate estimates is played by the constant @c in the so-called strengthened Cauchy-Bunyakowski-Schwarz (CBS) inequality, associated with the angle between the two subspaces of the splitting. More precisely, the value of the upper bound for @c@?(0,1) is a part of the construction of various multilevel extensions of the related two-level methods. In this paper algebraic two-level and multilevel preconditioning algorithms for second-order elliptic boundary value problems are constructed, where the discretization is done using Rannacher-Turek non-conforming rotated bilinear finite elements on quadrilaterals. An important point to make is that in this case the finite element spaces corresponding to two successive levels of mesh refinement are not nested in general. To handle this, a proper two-level basis is required to enable us to fit the general framework for the construction of two-level preconditioners for conforming finite elements and to generalize the method to the multilevel case. The proposed variants of the hierarchical two-level basis are first introduced in a rather general setting. Then, the parameters involved are studied and optimized. The major contribution of the paper is the derived estimates of the constant @c in the strengthened CBS inequality which is shown to allow the efficient multilevel extension of the related two-level preconditioners. Representative numerical tests well illustrate the optimal complexity of the resulting iterative solver.