English abstract
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.
