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.
"Finite element schemes based on energy-stable approximation for two-fluid flow problems with surface tension." Hokkaido Math. J. 36 (4) 875 - 890, November 2007. https://doi.org/10.14492/hokmj/1272848038