  The eXtended Finite Element Method (XFEM) is an approach for solving problems
with non-smooth solutions. In the XFEM, the approximate solution is locally
enriched to capture discontinuities without requiring a mesh which conforms to
the geometric features. One drawback of the XFEM is that an ill-conditioned
system of equations results when the ratio of volumes on either side of the
interface in an element is small. In this paper, to avoid this
ill-conditioning, a simple and efficient scheme based on a geometric
preconditioner and constraining degrees of freedom to zero for small
intersections is proposed. This geometric preconditioner is computed from the
nodal basis functions, and therefore may be constructed prior to building the
system of equations. This feature and the low-cost of constructing the
preconditioning matrix makes it well suited for nonlinear problems with fixed
and moving interfaces.
