The subject of this thesis is the derivation and analysis of numerical approximations of multi-component, multi-phase field systems. Recent approximations of solutions to such models are mostly based on explicit time stepping schemes and require the computation of many time steps. Implicit methods exhibit inherent numerical challenges, in particular due to the non-smoothness of the underlying energy functionals.

Our focus lies on the derivation of numerical approximations within the thermodynamically consistent context with high efficiency and robustness. We aim to exploit the special mathematical structure of the model and the underlying thermodynamics without introducing additional regularizations.

We introduce the thermodynamic and multi-phase setting in chapter 2 and continue by motivating and presenting a thermodynamically consistent multi-component, multi-phase field model in chapter 3. Based on Rothe’s method, we obtain a semi-discretization allowing for adaptive meshes in chapter 4 and the implicit problems are analyzed. In chapter 5, a full discretization with adaptive finite elements based on hierarchical a posteriori error estimation is set up. We transition to a purely algebraic formulation and present the iterative approximation of solutions with a nonsmooth Schur–Newton multigrid approach in chapter 6. Finally, in chapter 7, we perform numerical experiments to underline the thermodynamical consistency and numerical efficiency of our method.