Analysis of an extended pressure finite element space for two-phase incompressible flows

We consider a standard model for incompressible two-phase flows in which a localized force at the interface describes the effect of surface tension. If a level set (or VOF) method is applied then the interface, which is implicitly given by the zero level of the level set function, is in general not aligned with the triangulation that is used in the discretization of the flow problem. This non-alignment causes severe difficulties w.r.t. the discretization of the localized surface tension force and the discretization of the flow variables. In cases with large surface tension forces the pressure has a large jump across the interface. In standard finite element spaces, due to the non-alignment, the functions are continuous across the interface and thus not appropriate for the approximation of the discontinuous pressure. In many simulations these effects cause large oscillations of the velocity close to the interface, so-called spurious velocities. In [6] it is shown that an extended finite element space (XFEM) is much better suited for the discretization of the pressure variable. In this paper we derive important properties of the XFEM space. We present (optimal) approximation error bounds and prove that the diagonally scaled mass matrix has a uniformly bounded spectral condition number. Results of numerical experiments are presented that illustrate properties of the XFEM space.