  We consider discrete Poisson interface problems resulting from linear unfitted finite elements, also called cut finite elements (CutFEM). Three of these unfitted finite element methods known from the literature are studied. All three methods rely on Nitsche s method to incorporate the interface conditions. The main topic of the paper is the development of a multigrid method, based on a novel prolongation operator for the unfitted finite element space and an interface smoother that is designed to yield robustness for large jumps in the diffusion coefficients. Numerical results are presented which illustrate efficiency of this multigrid method and demonstrate its robustness properties with respect to variation of the mesh size, location of the interface and contrast in the diffusion coefficients. 
A Multigrid Method for Unfitted Finite Element Discretizations of   Elliptic Interface Problems