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, P. Hansbo, Comput. Methods Appl. Mech. Engrg. 191 (20002)], for the purely elliptic stationary case, extended finite elements (XFEM) are combined with a Nitsche-type of method, and optimal error bounds are derived. These results were extended to the unsteady case in [A. Reusken, T. Nguyen, J. Fourier Anal. Appl. 15 (2009)]. 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 method. 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 method is presented that results in optimal a-priori discretization error bounds. AMS subject classification. 65N12, 65N30