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The role of the strengthened Cauchy-Buniakowskii-Schwarz inequality in multilevel methods. (English) Zbl 0737.65026

This is a survey paper where the role of the strengthened Cauchy inequality in the multilevel technique is summarized.
In the first part various facts about the angle between two subspaces \(H\) and \(V\) of \(\mathbb{R}^ n\) are presented using purely algebraic approach. The authors consider various cases of symmetric semidefinite matrices \(M\) split into \(2\times 2\) block structure (induced by the subspaces \(V\) and \(H\)): \(M=\left({B\atop C}{C^ T\atop A}\right)\) where \(B\) is invertible and \(A\) semidefinite; or \(B\) and \(A\) invertible, etc.
The next part is devoted to the application of the strengthened Cauchy inequality to construct and investigate two-level and multilevel preconditioners for finite element approximations of second order symmetric elliptic problems. In particular, the following result is proved: the angle between two hierarchical finite element spaces is uniformly (with respect to the mesh size) bounded by a constant less than 1.

MSC:

65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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