A stable XFEM formulation for multi-phase problems enforcing the accuracy of the fluxes through Lagrange multipliers

When applied to diffusion problems in a multiphase setup, the popular XFEM strategy suffers from an inaccurate representation of the local fluxes in the vicinity of the interface. The XFEM enrichment improves the global quality of the solution but it is not enforcing any local feature to the fluxes. Thus, the resulting numerical fluxes in the vicinity of the interface are not realistic, in particular when conductivity ratios between the different phases are very high. This paper introduces an additional restriction to the XFEM formulation aiming at properly reproducing the features of the local fluxes in the transition zone. This restriction is implemented through Lagrange multipliers, and the stability of the resulting mixed formulation is tested satisfactorily through the Chapelle-Bathe numerical procedure. Several examples are presented and the solutions obtained show a spectacular improvement with respect to the standard XFEM.