The thesis is composed of three chapters which provide the convergence results for finite volume (FV) methods.

In the first chapter, we consider finite volume methods based on a non-uniform grid and present an approach to the numerical analysis of FVs (rectangular elements). We compare the FV methods with finite difference methods based on a related grid and define an isomorphism between the FV spaces and the FD spaces based on the same grids. Using this isomorphism, we then prove that the available convergence results in FD approximations lead to convergence results for FV approximations. The mentioned results are done for both dimension one and dimension two and are based on the external space approximation framework of Lions and Céa.

In the second chapter, we consider the applications of the convergence results developed in chapter one. We discretize a general two-dimensional Dirichlet problem by FVs and prove the convergence of its FV schemes. We also prove the convergence results of FV schemes for some parabolic evolution problems in space dimension two.

In the last chapter, we consider an application not covered by the theory in chapter one. We discretize the 2D linearized Shallow Water equations using FVs in space and an Implicit Euler scheme in time and then prove stability and convergence results for these schemes.