Finite element schemes based on energy-stable approximation for two-fluid flow problems with surface tension

For two-fluid flow problems with surface tension we present finite element schemes based on energy-stable approximation. In the case of no surface tension, those schemes are unconditionally stable in the energy-sense. When there exists surface tension, they are proved to be stable if a quantity remains bounded in the computation. Some numerical results of rising bubble problems show the robustness and applicability of these schemes.