We consider the evolution of moving contact lines in two-phase flows by using a sharp interface approach. The dynamic equations are governed by the standard incompressible Naiver-Stokes equations in the bulk coupled with the interface jump condition across the fluid interface, boundary conditions for the slip velocity on the substrate and the dynamic contact angle condition at contact lines. We propose an efficient finite element method for the problem with the moving fitted mesh such that the discretization of the fluid interface is fitted into the bulk triangular mesh. The capillary curvature term for the surface tension is treated by a variational formulation coupled with the dynamic contact angle condition. This method can be shown to be unconditionally stable in each time step. Extensive numerical results are shown to demonstrate the applicability, accuracy and high efficiency of the proposed numerical method.
