Explicit error bounds for an adaptive finite volume scheme

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.