Optimal preconditioners for Nitsche-XFEM discretizations of interface problems

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.