The shape and velocity of a sliding droplet are computed by solving the Navier-Stokes equation with free interface boundary conditions. The Galerkin finite element methodis implemented in a 2D computation domain discretized using an unstructured mesh withtriangular elements. The mesh is refined recursively at the corners (contact points). Thestationary sliding velocity is found to be strongly dependent on grid refinement, which isa consequence of the contact line singularity resolved through the effective slip across thefinite elements adjacent to the contact point. For small droplets, this dependence is wellapproximated by a theoretical estimates obtained using multiscale expansion and matchingtechnique in lubrication approximation, where the corner element size is used as a microscaleparameter. For larger droplets, the shape is also dependent on grid refinement. This questions the validity of numerous computations of flows with moving contact line where gridsare invariably much more coarse than molecular scales on which the singularity is resolved. Itis suggested that extrapolation to molecular scales should be used to obtain realistic results.