Flux-free Finite Element Method with Lagrange Multipliers for Two-fluid Flows

In this paper we consider the flux-free finite element method based on the Eulerian framework for immiscible incompressible two-fluid flows, which is defined so as to preserve the mass of each fluid. This method is derived from the variational formulation including the flux-free constraint for the Navier–Stokes equations by the Lagrange multiplier technique. Focusing on the stationary problem, we prove the well-posedness of the finite element solution by a discrete inf-sup condition and show basic error estimates. Moreover we also show the stability of the fractional-step projection finite element scheme for the non-stationary problem. Finally, we give some numerical results to validate our method.