Abstract. 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, be a convex polygonal domain that contains two different immiscible incompressible phases. The (in general time dependent) subdomains containing the two phases are denoted by Ω 1 , Ω 2 , withΩ =Ω 1 ∪Ω 2 . The interface Γ :=Ω 1 ∩Ω 2 is assumed to be sufficiently smooth. A model example is a (rising) droplet in a flow field. The fluid dynamics in such a flow problem is usually modeled by the incompressible Navier-Stokes equations combined with suitable conditions at the interface which describe the effect of surface tension. For this model we refer to the literature, e.g. [3,8,15,21,9]. By w we denote the velocity field resulting from these Navier-Stokes equations. We assume that div w = 0 holds. Furthermore, we assume that the transport of the interface is determined by this velocity field, in the sense that V Γ = w · n holds, where V Γ is the normal velocity of the interface and n denotes the unit normal at Γ pointing from Ω 1 into Ω 2 . In this paper we restrict ourselves to the case of a stationary interface, i.e., we assume w · n = 0. This case is (much) easier to handle than the case of an non-stationary interface Γ = Γ(t). We restrict to this simpler case because even for that the issue of stabilization of the Nitsche-XFEM method for convection-dominated transport problems has not been investigated, yet. The case of a non-stationary interface will be studied in a forthcoming paper. We comment on this further in Remark 6 at the end of the paper. We consider a model which describes the transport of a dissolved species in a two-phase