Analysis of Time Discretization Methods for a Two–Phase Stokes Equation with Surface Tension

In two-phase incompressible flow problems surface tension effects often play a key role. Due to surface tension the pressure is discontinuous across the interface. In interface capturing methods the grids are typically not aligned to the interface and thus in problems with an evolving interface time dependent pressure spaces should be used. Hence, a method of lines approach is not very suitable for this problem class. We consider a Rothe method with an implicit Euler or a Crank-Nicolson time discretization method. The order of convergence of these methods is not clear, since the surface tension force results in a right-hand side functional in the momentum equation with poor regularity properties. These regularity properties are such that for the Crank-Nicolson method one can not apply error analyses known in the literature. In this paper, for a simplified non-stationary Stokes problem a convergence analysis is presented. The analysis leads to optimal order error bounds. For the Crank-Nicolson method the error analysis uses a norm that is weaker that the L2-norm. Results of numerical experiments are shown that confirm the analysis.