Two-frequency decomposition. (English) Zbl 1066.65049

Considering large linear systems with special structure, the two authors unify their earlier separate work on filtering decompositions and propose a two-frequency decomposition (for block-tridiagonal, symmetric positive definite matrices) in which the filter condition is satisfied only approximately. This makes it easier to construct the filter but more difficult to investigate the rate of convergence of the iteration when the decomposition is used to define a splitting of the matrix or as preconditioner in the frames of the conjugate gradient method.
A great part of the paper is devoted to an estimate of the norm of the iteration operator in dependence of two parameters and to the exact solution of the corresponding optimization problem. Numerical results are presented for the model problem of the Poisson equation with Dirichlet boundary conditions in the unit square where the former estimate with optimal parameters guarantees a convergence rate of \(1-O(h^{2/3}\).


65F35 Numerical computation of matrix norms, conditioning, scaling
65F50 Computational methods for sparse matrices
65F10 Iterative numerical methods for linear systems
65N06 Finite difference methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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