Two methods to impose Dirichlet boundary conditions on an embedded interface are presented. The first one uses XFEM with the use of Lagrange multipliers and a stabilization term. The second one uses anisotropic adaptation close to the interface. The two methods are compared on benchmark problems.
Bricteux, G., & Toulorge, T. (2012). Alternative methods to represent embedded interfaces in a mesh. Berlin PUM Workshop 2012, Berlin. https://hdl.handle.net/2078.5/39560