We give an a posteriori error estimator for nonconforming finite element approximations of diffusionreaction and Stokes problems, which relies on the solution of local problems on stars. It is proved to be equivalent to the energy error up to a data oscillation, without requiring Helmholtz decomposition of the error nor saturation assumption. Numerical experiments illustrate the good behavior and efficiency of this estimator for generic elliptic problems.
International audience
We derive and analyze an a posteriori error estimator for nonconforming finite element approximation for the quasi-Stokes problem, which is based on the solution of local problem on stars with low cost computation, this indicator is equivalent to the energy error norm up to data oscillation, neither saturation assumption nor comparison with residual estimator are made.
A new technique for the implementation of cell-centered finite volume schemes is proposed. It is based on an equivalence between these schemes and the non-conforming Crouzeix-Raviart finite element approximation. Though the implementation somewhat increases the number of unknowns, it brings two significant advantages that make it attractive. On the one hand the resulting formulation is well-defined on general unstructured meshes, with no restriction on (or ad-hoc treatment of) obtuse angles. On the other hand, it automatically provides a very effective, asymptotically exact, error estimator. An adaptive finite volume method, based on this estimator, is proposed and tested. It turns out to be remarkable for its exactness, even in coarse meshes.
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