Characteristics/finite element analysis for two incompressible fluid flows with surface tension using level set method

Abstract

In this paper, we present and analyze a finite element level set method based on the method of characteristics for two phase flow. Surface tension effects are taken into account by the CSF approach. We first write the variational formulation of the problem and investigate its well-posedness. Next, for the discretization, a first order method of characteristics approach for the evolution of the level set function and for the material derivative of the velocity is used. The velocity and pressure unknowns are discretized by P2−P1 Taylor–Hood finite elements. Then, in each time step, the interface transport is decoupled from the Navier–Stokes equations. Well–posedness results for subproblems in this decoupled discrete problem are derived. Furthermore, under high regularity assumptions, we state error estimates for our scheme. Ultimately, three computational examples illustrate the performance of the proposed method.