An energy diminishing arbitrary Lagrangian–Eulerian finite element method for two-phase Navier–Stokes flow

Structure-preserving discretizations of two-phase Navier–Stokes flow using fitted and unfitted approaches

“…We note that (4.20) is similar to [21, (2.26)-(2.29)] except that here we employ the semi-implicit approximation of the unit normal from (3.12) instead of ν m for the stability and volume conservation. In addition, the scheme from [21] is based on a formulation with a divergence free ALE mesh velocity, see Remark 4.1. In the construction of the discrete ALE mappings, the discrete displacement is then solved via a Stokes equation, instead of the elastic equation we consider in (4.15).…”

“…For example, the curvature of the interface, denoted by , can be accurately computed with the help of the identity [20,22] ν = s id, (1.1) where id is the identity function, ν is the unit normal to the interface, s = ∇ s • ∇ s is the Laplace-Beltrami operator on the interface with ∇ s being the surface gradient. The identity (1.1) was initially used for the computation of mean curvature flow in [22], and then has been generalized to approximate the surface tension force in the context of two-phase flow, e.g., [3,4,9,12,14,21,27,52,53].…”

“…In purely Lagrangian schemes, on the other hand, the bulk mesh needs to deform according to the fluid velocity, and this often leads to large distortions of the mesh. A possible way to overcome these drawbacks is the so-called arbitrary Lagrangian-Eulerian (ALE) approach, e.g., [4,7,21,26,28,33], where the equations are formulated in a moving frame of reference, and the corresponding reference velocity is independent from the fluid velocity, and in this sense it is "arbitrary".…”

“…Moreover, applications to the two-phase Navier-Stokes flow with moving fitted finite element methods were also investigated in [4] in both Eulerian and ALE approaches, but in general no stability bound was available. Recently, by assuming that the ALE frame velocity satisfies the divergence-free condition, Duan, Li and Yang proposed an ALE weak formulation, which leads to an energy-diminishing scheme on the discrete level [21]. In addition to these energy stable approximations, numerical approximations that maintain the volume preservation of the two phases are also desirable.…”

“…The introduced scheme is based on suitable discretizations of a novel nonconservative ALE weak formulation. We argue that in terms of the treatment of the inertia term, our ALE method is similar to the work in [21], but here we allow for a more flexible mesh velocity of the ALE frame, which helps to guarantee the quality of the bulk mesh.…”