In this thesis, we are dealing with the solution of nonlinear, coupled partial differential
equations. These are challenging problems with a multitude of applications, e.g., the
flow of air around a plane, or the behavior of blood flow in arteries. Such examples are
categorized as fluid-structure interaction problems, which are one of the main models
treated in this work. Furthermore, we investigate the numerical simulation of fractures
in brittle materials such as concrete or glass. Fractures are usually a lower dimensional
phenomenon, i.e., a crack in a sheet of glass is just a thin line. This makes its treatment
by standard methods more challenging. In this work, we use the phase-field approach
to fracture propagation. This strategy aims to extend a thin crack in all directions,
eventually getting rid of the lower dimensionality. The final problem can be written
as an energy minimization problem, which then reduces to the solution of a nonlinear,
coupled partial differential equation. Furthermore, a fracture should not be able to heal
itself. Mathematically, this leads to a variational inequality, which requires specialized
solvers to deal with these constraints.
We focus on the parallel solution of the linear systems arising after discretization and
linearization of the differential equations. We use a matrix-free approach to overcome the
huge memory requirements of standard, sparse matrix based methods, in particular for
high-order polynomial shape functions. Since the matrix itself is not available, geometric
multigrid methods are one of the few viable options to solve these systems of equations.
Numerical experiments illustrate the applicability and performance of matrix-free methods
for challenging problems like fracture propagation and fluid-structure interaction.
The implemented matrix-free solver is compared with available matrix based ones in
terms of performance and robustness.