Céa-type quasi-optimality and convergence rates for (Adaptive) vertex-centered FVM

Abstract

For a general second order linear elliptic PDE, we show a generalized Céa lemma for a vertex-centered finite volume method (FVM). The latter implies, in particular, a comparison result between the solutions of FVM and the finite element method (FEM). Furthermore, for a symmetric PDE, i.e., no convection is present, we prove linear convergence with generically optimal algebraic rates for an adaptive FVM algorithm.