Eulerian formulations of problems with interfaces avoid the subtleties of tracking and remeshing, but do they complicate solution of the discrete equations, relative to domain decomposition methods that respect the interface? We consider two different interface problems – one involving cracking and one involving phase separation. Crack problems can be formulated by extended finite element methods (XFEM), in which discontinuous fields are represented via special degrees of freedom. These DOFs are not properly handled in a typical AMG coarsening process, which leads to slow convergence. We propose a Schwarz approach that retains AMG advantages on the standard DOFs and avoids coarsening the enriched DOFs. This strategy allows reasonably mesh-independent convergence rates, though the convergence degradation of the (lower dimensional) set of crack DOFs remains to be addressed. Phase separation problems can be formulated by the Cahn-Hilliard approach, in which the level set of a continuous Eulerian field demarcates the phases. Here, scalable preconditioners follow naturally, once the subtlety of the temporal discretization is sorted out. The first project is joint with R. Tuminaro and H. Waisman and the second with X.-C. Cai and C. Yang.