Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients
Résumé
We study in this paper a posteriori error estimates for H1-conforming numerical approximations of diffusion problems with a scalar, piecewise constant, and arbitrarily discontinuous diffusion coefficient. We derive estimators for the energy norm and the dual norm of the residual which give a guaranteed global upper bound in the sense that they feature no undetermined constants. (Local) lower bounds, up to constants independent of the diffusion coefficient, are also derived. In particular, no condition on the diffusion coefficient like its monotonous increasing along paths around mesh vertices is imposed, whence the present results are fully robust and include also the cases with singular solutions. For the energy error setting, the key requirement turns out to be that the diffusion coefficient is piecewise constant on dual cells associated with the vertices of an original simplicial mesh and that harmonic averaging is used in the scheme. This is the usual case, e.g., for the cell-centered finite volume method, included in our analysis as well as the vertex-centered finite volume, finite difference, and continuous piecewise linear finite element ones. For the dual norm setting, no such a requirement is necessary. Our estimates are based on H(div)-conforming flux reconstruction obtained thanks to the local conservativity of all the studied methods on the dual grids, which we recall in the paper, together with their mutual relations. Numerical experiments confirm the guaranteed upper bound, full robustness, and excellent efficiency of the derived estimators.
Origine : Fichiers produits par l'(les) auteur(s)