Hierarchical error estimates for finite volume approximation solution of elliptic equations

We propose hierarchical error estimates for the vertex-centered finite volume approximation solution of elliptic equations. The reliability and efficiency of hierarchical estimates are analyzed by showing that the H1-norm of the finite volume approximate error are equivalent to our hierarchical estimator up to some oscillation terms which are supposed to be high order terms. Moreover, we show that the small oscillation implies the saturation property, as what has been proved for finite elements approximation in Dörfler and Nochetto (2002) [16]. Numerical experiments confirm our theoretic findings.