An extended finite element method for a diffuse-interface model

This work presents an extended finite element method (XFEM) for a diffuse-interface model, which describes interfacial phenomena of multi-phase flow. The diffuse interface has a non-zero thickness over which the phase field variable changes continuously. The diffuse-interface thickness is typically very small compared to the observed domain size, resulting in a high-gradient solution. In this work, the finite element approximation of the phase field variable is locally enriched with a tangent hyperbolic function which characterizes a high-gradient solution of the diffuse-interface model. We study a one-dimensional advection and two diffusion problems, and demonstrate the remarkable improvement of the solution by the local enrichment.