The finite element method for diffusion problems with two phases: XFEM and XFEM+

Obtaining precise solutions with the finite element method (FEM) for diffusive problems with multiple phases entails a high computational cost. For this reason, the extended finite element method (XFEM) has become the usual tool for the analysis of this type of problems. However, when XFEM is applied to two-phase problems with very different conductivities, inaccurate representations of the flows in the vicinity of the interface are obtained. To alleviate this deficiency of the XFEM, additional restrictions must be added to the method, originating a modification of it, known as XFEM +. Since XFEM + is relatively recent and little known and implemented at the time of designing applications for real problems of engineering and science, this paper presents its theoretical development, as well as the numerical results obtained for 2D problems, compared to FEM and XFEM. A contribution of this work is given by the implementation of the methods for quadrilateral elements. The numerical results obtained for this type of elements, and different interfaces, show the remarkable improvement of the XFEM when increasing the order of interpolation.