  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.
