Optimal convergence of the arbitrary Lagrangian-Eulerian interface tracking method for two-phase Navier-Stokes flow without surface tension

Optimal-order convergence in the H¹ norm is proved for an arbitrary Lagrangian–Eulerian (ALE) interfacetracking finite element method (FEM) for the sharp interface model of two-phase Navier–Stokes flowwithout surface tension, using high-order curved evolving mesh. In this method, the interfacial mesh pointsmove with the fluid’s velocity to track the sharp interface between two phases of the fluid, and the interiormesh points move according to a harmonic extension of the interface velocity. The error of the semidiscreteALE interface tracking FEM is shown to be O(h^k) in the L^∞ (0, T; H¹ (Ω) ) norm for the Taylor–Hoodfinite elements of degree k ≥  2. This high-order convergence is achieved by utilizing the piecewisesmoothness of the solution on each subdomain occupied by one phase of the fluid, relying on a low globalregularity on the entire moving domain. Numerical experiments illustrate and complement the theoreticalresults.

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