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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 $L^2(\Gamma)$.

We prove convergence for the nonoverlapping Robin-Robin method applied to nonlinear elliptic equations with a p-structure, including degenerate diffusion equations governed by the p-Laplacian. This nonoverlapping domain decomposition is commonly encountered when discretizing elliptic equations, as it enables the usage of parallel and distributed hardware. Convergence has been derived in various linear contexts, but little has been proven for nonlinear equations. Hence, we develop a new theory for nonlinear Steklov-Poincaré operators based on the p-structure and the L^p-generalization of the Lions-Magenes spaces. This framework allows the reformulation of the Robin-Robin method into a Peaceman-Rachford splitting on the interfaces of the subdomains, and the convergence analysis then follows by employing elements of the abstract theory for monotone operators. The analysis is performed on Lipschitz domains and without restrictive regularity assumptions on the solutions.

We consider a fully implicit formulation for two-phase flow in a porous medium with capillarity, gravity, and compressibility in three dimensions. The method is implicit in time and uses the multiscale mortar mixed finite element method for a spatial discretization in a nonoverlapping domain decomposition context. The interface conditions between subdomains are enforced in terms of Lagrange multiplier variables defined on a mortar space. The novel approach in this work is to linearize the coupled system of subdomain and mortar variables simultaneously to form a global Jacobian. This algorithm is shown to be more efficient and robust compared to previous algorithms that relied on two separate nested linearizations of subdomain and interface variables. We also examine various upwinding methods for accurate integration of phase mobility terms near subdomain interfaces. Numerical tests illustrate the computational benefits of this scheme.