Finite element discretization error analysis of a surface tension force in two-phase incompressible flows

We consider a standard model for a stationary two-phase incompressible flow with surface tension. In the variational formulation of the model a linear functional which describes the surface tension force occurs. This functional depends on the location and the curvature of the interface. In a finite element discretization method the functional has to be approximated. For an approximation method based on a Laplace-Beltrami representation of the curvature we derive sharp bounds for the approximation error. A new modified approximation method with a significantly smaller error is introduced.