This manuscript is a summary of the research works that I have done since my PhD Thesis. It is composed of two main chapters of non-equal sizes. The first and shorter one summaries my work in the context of elliptic problems and incompressible flows. The second and larger chapter presents more precisely my work dealing with hyperbolic problems and compressible flows. Actually, there is no direct link between these two topics, but the influence of the techniques that I used for elliptic problems is not negligible with respect to some of the works presented in the second chapter. For instance, the semi-Lagrangian finite-volume solver designed to treat curvilinear cells is based on a finite-element approximation of the fluxes.