The project is concerned with the development of innovative enriched Galerkin methods for the reliable simulation of pressure-driven fracture problems. Within this scope the project will consider the soundmodeling of pressure driven fractures in(quasi-)incompressible materials for quasi-static and dynamicfracture-propagation.Emphasis will be given to the analysis of the incompressiblelimit by combining mixed formulations together withrecent pressure robust discretizations for the Stokes-equation. This will be complemented by designing convergent adaptive mesh-refinement schemes based on new efficient error estimators for the variational inequality associated with the fracture irreversibility.