On time integration in the XFEM

The extended finite element method (XFEM) is often used in applications that involve moving interfaces. Examples are the propagation of cracks or the movement of interfaces in two-phase problems. This work focuses on time integration in the XFEM. The performance of the discontinuous Galerkin method in time (space-time finite elements (FEs)) and time-stepping schemes are analyzed by convergence studies for different model problems. It is shown that space-time FE achieve optimal convergence rates. Special care is required for time stepping in the XFEM due to the time dependence of the enrichment functions. In each time step, the enrichment functions have to be evaluated at different time levels. This has important consequences in the quadrature used for the integration of the weak form. A time-stepping scheme that leads to optimal or only slightly sub-optimal convergence rates is systematically constructed in this work.