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A multi-grid algorithm for mixed problems with penalty. (English) Zbl 0712.73106
The author develops a multigrid algorithm for the finite element approximation of mixed problems with penalty by the MINI element. Previously the Mindlin-Reissner plate model was applied in a mixed method with penalty [the author, A multigrid algorithm for mixed problems with penalty. Bochum: Univ. Bochum, Abt. f. Math., Diss. (1989; Zbl 0696.73043)] giving an optimal convergence rate uniformly with respect to the plate thickness. It enabled to avoid spurious phenomena like locking. Here it is proved that the convergence rate in a given multigrid algorithm is bounded away from 1 independently of the mesh size and of the thickness parameter. This aim is obtained by using smoothing procedures with an appropriate t-dependence. The convergence also holds when another finite element used in fluid mechanics is considered.
This mathematical paper is addressed to scientists working in the theory of the finite element method and for theoreticians who develop numerical models for practical use.
Reviewer: Cz.I.Bajer

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
74S05 Finite element methods applied to problems in solid mechanics
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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