  In this paper we consider a class of unfitted finite element methods for
scalar elliptic problems. These so-called CutFEM methods use standard finite
element spaces on a fixed unfitted triangulation combined with the Nitsche
technique and a ghost penalty stabilization. As a model problem we consider the
application of such a method to the Poisson interface problem. We introduce and
analyze a new class of preconditioners that is based on a subspace
decomposition approach. The unfitted finite element space is split into two
subspaces, where one subspace is the standard finite element space associated
to the background mesh and the second subspace is spanned by all cut basis
functions corresponding to nodes on the cut elements. We will show that this
splitting is stable, uniformly in the discretization parameter and in the
location of the interface in the triangulation. Based on this we introduce an
efficient preconditioner that is uniformly spectrally equivalent to the
stiffness matrix. Using a similar splitting, it is shown that the same
preconditioning approach can also be applied to a fictitious domain CutFEM
discretization of the Poisson equation. Results of numerical experiments are
included that illustrate optimality of such preconditioners for the Poisson
interface problem and the Poisson fictitious domain problem.
