# Parallel Multigrid Solvers forNonlinear Coupled Field Problems

## DI Daniel Jodlbauer

**March 2, 2021, 2:30 p.m. ZOOM**

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 propose 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.