Residual flux-based a posteriori error estimates for finite volume discretizations of inhomogeneous, anisotropic, and convection-dominated problems
Résumé
We derive in this paper a posteriori error estimates for finite volume discretizations of convection-diffusion-reaction equations, particularly focusing on the cases of inhomogeneous and anisotropic diffusion-dispersion tensors and of convection dominance. Our estimates are valid for any cell-centered finite volume scheme on a mesh consisting of simplices or rectangular parallelepipeds and extensions to general polygonal/polyhedral meshes including the nonmatching ones are provided as well. The estimates are established in the energy (semi-)norm for a locally postprocessed approximate solution preserving the finite volume fluxes and are of residual type. We prove that they are reliable (yield a global upper bound on the error), efficient (give a local lower bound on the error times an efficiency constant), and robust in the sense that the efficiency constant only depends on local variations in the coefficients and becomes optimal as the local Péclet number gets sufficiently small. The derived estimators are fully computable (all occurring constants are evaluated explicitly), so that they can serve both as indicators for adaptive refinement or for the actual control of the error. Numerical experiments confirm their accuracy.