  We consider the dynamics of two-phase fluids, in particular the moving
contact line, on a solid substrate. The dynamics are governed by the
sharp-interface model consisting of the incompressible Navier-Stokes\slash
Stokes equations with the classical interface conditions, the Navier boundary
condition for the slip velocity along the wall and a contact line condition
which relates the dynamic contact angle of the interface to the contact line
velocity. We propose an efficient numerical method for the model. The method
combines a finite element method for the Navier-Stokes/Stokes equations on a
moving mesh with a parametric finite element method for the dynamics of the
fluid interface. The contact line condition is formulated as a time-dependent
Robin-type of boundary condition for the interface so it is naturally imposed
in the weak form of the contact line model. For the Navier-Stokes equations,
the numerical scheme obeys a similar energy law as in the continuum model but
up to an error due to the interpolation of numerical solutions on the moving
mesh. In contrast, for Stokes flows, the interpolation is not needed so we can
prove the global unconditional stability of the numerical method in terms of
the energy. Numerical examples are presented to demonstrate the convergence and
accuracy of the numerical methods.
