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Finite volume schemes: hierarchical a posteriori error estimation through mixed finite elements. Numerical solution of non-linear elasticity problems.

Schema volumes finis : Estimation d'erreur a posteriori hierarchique par elements finis mixtes. Resolution de problemes d'elasticite non-linearie

Abstract : Part one is cocerned with numerical analysis. starting from a mixed finite element interpretation of basic finite volume (F.V.) schemes, a posteriori error estimation is analysed in the hirarchy of Raviart-Thomas elements. An explicit compatible estimator is given for these F.V. schemes.
Part two introduces a family of F.V.. schemes of finite differences type, for a rectangular mesh and a more general structured one. Numerical experiments, for model problems, show that the precision order of the theoretical analysis may be reached.
Part three presents the application of the F.V. schemes to the numerical simulation of the deformations of a ruber bloc containing a finite crack. This corresponds to large deformations of a compressible hyperelastic material. The numerical experiments correspond to a constitutive law of Saint-Venant-Kirchhoff type. The results give the deformations and different stress tensors, and first tests for qusi-incompressibily and damage simulations.