  In the past decade, a combination of unfitted finite elements (or XFEM) with
the Nitsche method has become a popular discretization method for elliptic
interface problems. This development started with the introduction and analysis
of this Nitsche-XFEM technique in the paper [A. Hansbo, P. Hansbo, Comput.
Methods Appl. Mech. Engrg. 191 (2002)]. In general, the resulting linear
systems have very large condition numbers, which depend not only on the mesh
size $h$, but also on how the interface intersects the mesh. This paper is
concerned with the design and analysis of optimal preconditioners for such
linear systems. We propose an additive subspace preconditioner which is optimal
in the sense that the resulting condition number is independent of the mesh
size $h$ and the interface position. We further show that already the simple
diagonal scaling of the stifness matrix results in a condition number that is
bounded by $ch^{-2}$, with a constant $c$ that does not depend on the location
of the interface. Both results are proven for the two-dimensional case. Results
of numerical experiments in two and three dimensions are presented, which
illustrate the quality of the preconditioner.
