Nitsche-XFEM with Streamline Diffusion Stabilization for a Two-Phase Mass Transport Problem

We consider an unsteady convection diffusion equation which models the transport of a dissolved species in two-phase incompressible flow problems. The so-called Henry interface condition leads to a jump condition for the concentration at the interface between the two phases. In [A. Hansbo and P. Hansbo, Comput. Methods Appl. Mech. Engrg., 191 (2002), pp. 5537--5552], for the purely elliptic stationary case, an extended finite element method (XFEM) is combined with a Nitsche-type method, and optimal error bounds are derived. These results were extended to the unsteady case in [A. Reusken and T. Nguyen, J. Fourier Anal. Appl., 15 (2009), pp. 663--683]. In the latter paper convection terms are also considered but assumed to be small. In many two-phase flow applications, however, convection is the dominant transport mechanism. Hence there is a need for a stable numerical method for the case of a convection dominated transport equation. In this paper we address this topic and study the streamline diffusion stabilization for the Nitsche-XFEM. The method is presented, and results of numerical experiments are given that indicate that this kind of stabilization is satisfactory for this problem class. Furthermore, a theoretical error analysis of the stabilized Nitsche-XFEM is presented that results in optimal a priori discretization error bounds.