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Two-level preconditioners for \(2m\)’th order elliptic finite element problems. (English) Zbl 0860.65120

Two-level techniques for constructing the preconditioners for the discrete systems which arise from the finite element approximation of second- and fourth-order elliptic boundary value problems are considered. A two-level preconditioner can be thought of as an additive version of the general two-level multigrid algorithm. Thus, the preconditioner for a problem on a given grid is constructed in terms of a solution or an approximate solution of an auxiliary problem on a related grid and a ‘smoother’ on the original grid.
Two abstract theorems are provided. Properties needed to apply these theorems are developed for general finite element approximation spaces. These results are then applied to the second-order and biharmonic Dirichlet problems. Also, the authors provide a theorem which shows that the natural additive two-level preconditioner gives rise to uniformly convergent iterations under a regularity and approximation assumption.

MSC:

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65F35 Numerical computation of matrix norms, conditioning, scaling
35J40 Boundary value problems for higher-order elliptic equations
31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
65F10 Iterative numerical methods for linear systems
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