In the lecture modern numerical methods for the discretization of interface problems are presented. Partial differential equations the solution of which has (weak) discontinuities across given interfaces require special care for their numerical treatment. The concept of extended finite element methods (XFEM) introduces finite element spaces which provide ansatz functions which are discontinuous across the interface. Based on this approach the XFEM is used for different applications. We will present interface problems arising in two-phase flow applications and discuss their numerical solution. The lecture includes

• the introduction of XFEM,
• the discussion of implementational aspects
• and their numerical analysis.