# Matrix-Free Multigrid for Phase-Field Fracture Problems

## DI Daniel Jodlbauer

**June 4, 2019, 4 p.m. S2 416-1**

Standard matrix-based FEM requires huge amounts of memory as the number of elements increases. This non neglectable drawback can be overcome by using matrix-free methods. Such methods do not require building and storing huge linear systems, instead they compute the necessary information on the fly. Hence, such approaches need far less memory than classical methods, which makes them the method of choice for very large problems. Without the matrix at hands, the number of available solvers is very limited, i.e., direct solvers and algebraic multigrid methods are no longer possible. A class of solvers that is very suitable in this case are geometric multigrid methods. These methods do not require explicit knowledge about the matrix entries and, thus, can be applied in a matrix-free fashion. In this talk, we present a framework for the matrix-free solution to a monolithic quasi-static phase-field fracture model. The equations of interest are nonlinear and need to satisfy a variational inequality. This imposes several challenges for the implementation, which will be discussed throughout the talk. Finally, several numerical examples are presented to show the applicability and parallel scalability of the matrix-free geometric multigrid solver.