Finite element approximation of the deterministic and the stochastic Cahn-Hilliard equation

Abstract: This thesis consists of three papers on numerical approximation of the Cahn-Hilliard equation. The main part of the work is concerned with the Cahn-Hilliard equation perturbed by noise, also known as the Cahn-Hilliard- Cook equation. In the first paper we consider the linearized Cahn-Hilliard-Cook equation and we discretize it in the spatial variables by a standard finite element method. Strong convergence estimates are proved under suitable assumptions on the covariance operator of the Wiener process, which is driving the equation. The analysis is set in a framework based on analytic semigroups. The main part of the work consists of detailed error bounds for the corresponding deterministic equation. Time discretization by the implicit Euler method is also considered. In the second paper we study the nonlinear Cahn-Hilliard-Cook equation. We show almost sure existence and regularity of solutions. We introduce spatial approximation by a standard finite element method and prove error estimates of optimal order on sets of probability arbitrarily close to 1. We also prove strong convergence without known rate. In the third paper the deterministic Cahn-Hilliard equation is considered. A posteriori error estimates are proved for a space-time Galerkin finite element method by using the methodology of dual weighted residuals. We also derive a weight-free a posteriori error estimate in which the weights are condensed into one global stability constant.