A finite element analysis of a linearized problem of the Navier-Stokes equations with surface tension

A finite element analysis is performed for a stationary linearized problem of the Navier-Stokes equations with surface tension. Since the surface tension brings about a second-order derivative of the velocity in the boundary condition, the velocity space is equipped with a stronger topology than in the conventional case. Under the strong topology, conditions of the uniform solvability and the approximation are verified on some pairs of finite element spaces for the velocity and the pressure. Thus an optimal error estimate is derived. Some numerical results are shown, which agree well with theoretical ones.