Discretizations of nonlinear dissipative evolution equations. Order and convergence

Abstract: The theme of this thesis is to study discretizations of nonlinear dissipative evolution equations, which arise in e.g. advection-diffusion-reaction processes. The convergence analysis is conducted by first considering an abstract time discretization of the problem, which enables a decoupling of the time and spatial approximations, and secondly by introducing the spatial discretization as an evolution on a finite dimensional space. For A-stable multistep methods and algebraically stable Runge-Kutta methods the very same global error bounds are obtained in this infinite dimensional setting as derived for stiff ODEs. Error bounds are also presented for full discretizations based on spatial Galerkin approximations. In contrast to earlier studies, our analysis is not relying on linearization procedures, but on the fully nonlinear framework of logarithmic Lipschitz constants and a generalization of the classical B-convergence theory.