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This thesis is concerned with the design and analysis of an interface fitted finite element strategy
for the simulation of free- and moving boundary problems with sharp interfaces. The presented
strategy falls into the class of interface tracking methods based on the Arbitrary Lagrangian-Eulerian (ALE) formulation. The main goal is to resolve the fundamental drawback of mesh
degeneration typical of moving mesh strategies while preserving all other benefits, in particular a
well-defined, accurate, and explicit representation of the moving geometry in space and time. This
representation allows for a straight-forward definition and implementation of problem tailored finite element spaces.
Parametric finite element spaces based on curved elements serve as the baseline of the presented
approach and are introduced and discussed in the context of a prototypical elliptic interface problem. A solution to the associated problem of generating admissible, interface fitted parametrizations is presented in terms of a variational mesh optimization approach. In this approach, a mesh
quality functional subject to an alignment constraint is minimized which leads to interface aligned,
optimal, and non-degenerate parametrizations.
For time-dependent problems, interface aligned parametrizations are constructed such that they
behave smoothly in time within each time slab, while a re-parametrization taking place in between
two successive time slabs is permitted. This strategy yields parametric finite element spaces that
are discontinuous in time. Consequently, a discontinuous Galerkin (dG(k)) approach in time is
used to construct fully discrete schemes. For the lowest order variant dG(0), an a priori error
analysis is conducted to show that the proposed strategy leads to convergence rates that are sub-
optimal by one order with respect to space in the worst case. A strategy to solve arising space-time
systems for higher-order dG(k) methods is proposed. This strategy is based on a transformation of
the space-time system into real block diagonal form, such that an efficient, preconditioned Schur
complement formulation for the arising 2 × 2 blocks can be used. It is shown that the preconditioned system possesses a condition number that is uniformly bounded by 2.
The last part of this thesis addresses the application of the strategy to free boundary problems. In
the context of a general flow problem in the presence of an interface force, governing equations
are introduced, a fully discrete space-time formulation is derived and an efficient first order in
time method is proposed. Two different application scenarios, a two-phase flow example with
surface tension, and a fluid-structure interaction problem are considered to validate and evaluate
the proposed approach, and to show its benefits and limitations.