In this thesis, we are concerned with the mathematical and numerical analysis of Schwarz Methods for Nonlinear Partial Differential Equations (NPDEs), in the context of non-matching grid. This kind of discretization is very interesting as it can be applied to solve many practical problems which cannot handled by global discretization. Our primary aim is to provide a mathematical sense to these computational methods by showing that the sub-problems resulting from the domain decomposition can be properly approximated and error estimates can be derived. More precisely, we consider both linear and nonlinear Schwarz iterations to approximate the solution of a class of semi linear PDEs, in the context of either finite element discretization on each sub-domain or hybrid discretization: finite elements on one sub-domain and finite difference on the other one. Our contribution is on both the theoretical and numerical simulation sides. The theoretical contribution of the thesis resides in both the uniform convergence of discretized linear Schwarz iterations to the true solution, error estimates between discretized nonlinear iterations and the true solution, and finite element error estimates for a Schwarz system associated with the class of PDEs under consideration. For that we developed two different methodologies encountered in modern applied mathematics: the method of upper and lower solutions and the method of sub solutions. The numerical simulation part consists of a variety of numerical experiments to support the theoretical findings. For that, we have adapted existing codes and developed new ones.