Convergence rate of a finite volume scheme for the linear convection-diffusion equation on locally refined meshes

We study a finite volume method, used to approximate the solution of the linear two dimensional convection diffusion equation, with mixed Dirichlet and Neumann boundary conditions, on Cartesian meshes refined by an automatic technique (which leads to meshes with hanging nodes). We propose an analysis through a discrete variational approach, in a discrete H-1 finite volume space. We actually prove the convergence of the scheme in a discrete H-1 norm, with an error estimate of order O(h) (on meshes of size h).