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Composite finite elements for the approximation of PDEs on domains with complicated micro-structures. (English) Zbl 0874.65086
The implementation of multigrid methods requires the use of hierarchical grids, from coarse to fine grids. Thus, these multigrid methods are well adapted to problems involving regular domains. In the case of domains with micro-structures, the basic multigrid method is no longer applicable since coarse meshes cannot take into account such micro-structures. The object of this paper is to introduce and to analyse mathematically finite element tools useful to extend such multigrid methods to partial differential equations (PDEs) formulated on domains with complicated micro-structures. A second forthcoming paper will present the effective implementation of such extensions.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
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