  In this paper we present a finite element method (FEM) for two-phase
incompressible flows with moving contact lines. We use a sharp interface
Navier-Stokes model for the bulk phase fluid dynamics. Surface tension forces,
including Marangoni forces and viscous interfacial effects, are modeled. For
describing the moving contact we consider a class of continuum models which
contains several special cases known from the literature. For the whole model,
describing bulk fluid dynamics, surface tension forces and contact line forces,
we derive a variational formulation and a corresponding energy estimate. For
handling the evolving interface numerically the level-set technique is applied.
The discontinuous pressure is accurately approximated by using a stabilized
extended finite element space (XFEM). We apply a Nitsche technique to weakly
impose the Navier slip conditions on the solid wall. A unified approach for
discretization of the (different types of) surface tension forces and contact
line forces is introduced. The numerical methods are first validated for
relatively simple test problems, namely a stationary spherical droplet in
contact with a flat wall and a spherical droplet on a flat wall that spreads or
contracts to a stationary form. A further validation is done for a two-phase
Couette flow with contact lines. To illustrate the robustness of our FEM we
also present results of simulations for a problem with a curved contact wall
and for a problem with more complicated contact line dynamics.
