A new redistancing method for piecewise polynomial finite element level set functions is introduced. The method directly computes the distance to the implicitly given discrete level set. Rigorous error bounds and numerical experiments are provided. Both show that, up to constants, the method is as good as the nodal interpolation of the computationally unavailable signed-distance function of the continuous level set.

We perform 3D incompressible two-phase flow simulations of rising droplets. Based on a similar 2D benchmark, a 3D benchmark configuration with two test cases is formulated in which we compare the flow solvers DROPS, NaSt3DGPF and OpenFOAM. All codes adopt different numerical techniques. We define several quantities of interest and investigate their temporal evolution in both test cases. For most benchmark variables we obtain a high level of agreement and establish reference data for other flow solvers.

We consider a time dependent Stokes problem that is motivated by two-phase incompressible flow problems with surface tension. The surface tension force results in a right-hand side functional in the momentum equation with poor regularity properties. As a strongly simplified model problem we treat a Stokes problem with a similar time dependent nonsmooth forcing term. We consider the implicit Euler and Crank-Nicolson methods for time discretization. The regularity properties of the data are such that for the Crank-Nicolson method one can not apply error analyses known in the literature. We present a convergence analysis leading to a second order error bound in a suitable negative norm that is weaker that the \(L^2\)-norm. Results of numerical experiments are shown that confirm the analysis.

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.

We consider a standard model for incompressible two‐phase flows in which a localized force at the interface describes the effect of surface tension. If a level set method is applied then the approximation of the interface is in general not aligned with the triangulation. This causes severe difficulties w.r.t. the discretization and often results in large spurious velocities. In this paper we reconsider a (modified) extended finite element method (XFEM), which in previous papers has been investigated for relatively simple two‐phase flow model problems, and apply it to a physically realistic levitated droplet problem. The results show that due to the extension of the standard FE space one obtains much better results in particular for large interface tension coefficients. Furthermore, a certain cut‐off technique results in better efficiency without sacrificing accuracy. Copyright © 2010 John Wiley & Sons, Ltd.