Construction and development of new schemes for elliptical or parabolic problems

In the first three chapters, we present a synthesis of the linear schemes that we have proposed since a decade for the discretization of scattering operators. The following chapters are devoted to the nonlinear schemes or corrections that we have proposed since 2005, allowing to respect the maximum principle for very long meshes or very anisotropic tensors. Indeed, for transport-chemistry models, or for two-phase models in porous media, it can be crucial to preserve the physical bounds. In chapter 4, we present the construction of schemes that are nonlinear variants of the so-called 'Diamond' method. The idea of the scheme is to compute the nonlinear combination of two flows on both sides of each edge. In Chapter 5, we construct nonlinear corrections applicable to a wide range of classical schemes (Finite Elements, Finite Volumes, etc.). We aim here at a double objective. First, we wish to keep the coercivity of the initial schemes. This property is important because it is useful to show the existence and convergence of nonlinear schemes. (author)