Convergence analysis of domain decomposition methods : Nonlinear elliptic and linear parabolic equations

Domain decomposition methods are widely used tools for solving partial differential equations in parallel. However, despite their long history, there is a lack of rigorous convergence theory for equations with non-symmetric differential operators. This includes both nonlinear elliptic equations and linear parabolic equations. The aim of this thesis is therefore twofold: First, to construct frameworks, based on new Steklov--Poincaré operators, that allow the study of nonoverlapping domain decomposition methods for nonlinear elliptic and linear parabolic equations. Second, to prove convergence of the Robin--Robin method using these frameworks. For the nonlinear elliptic case, this involves studying $L^p$-variants of the Lions--Magenes space. In the parabolic case, we use a variational formulation based on fractional time-regularity. The analysis is performed with weak requirements on the spatial domain, where we only assume that the domains have Lipschitz regularity, and for the solutions to the equations, where we assume that their normal derivatives over the interface is in $L^2(\Gamma)$.