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C. Erath, D. Praetorius:
"Céa-type quasi-optimality and convergence rates for (adaptive) vertex-centered FVM";
in: "Finite Volumes for Complex Applications VIII - Methods and Theoretical Aspects. FVCA 2017.", Springer, Wien, 2017, ISBN: 978-3-319-57396-0, 215 - 223.

For a general second order linear elliptic PDE, we show a generalized Cea 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.

finite volume method, Cea-type quasi-optimality, a posteriori error es-timators, adaptive algorithm, local mesh-refinement, optimal convergence rates

http://dx.doi.org/10.1007/978-3-319-57397-7_14