Nonconforming mixed finite element method for the diffuse interface model of two-phase fluid flows

Abstract

In this work, we consider a nonconforming mixed finite element method (MFEM) for a parabolic system consisting of the Cahn-Hilliard equations and the Navier-Stokes equations, which describes a diffuse interface model for the flow of two incompressible and immiscible fluids. We focus on the superconvergence analysis for the corresponding first order convex splitting scheme, where the bilinear element is used to approximate the Cahn-Hilliard equation in space, and the constrained nonconforming rotated (CNR) Q1 element together with the Q0 constant element to the Navier-Stokes equations. By use of the high accuracy properties of the elements and some specific techniques, the superclose results of order O(h2+τ) for the concentration ϕ and the chemical potential ω in the H1-norm, the velocity u in the broken H1-norm and the pressure p in L2-norm are strictly proved. Here, h is the subdivision parameter and τ, the time step. It is worth to say that we derive a new high accuracy character through the integral identity technique for the chosen elements, which plays a quite significant role in getting the final error results. Furthermore, the global superconvergence results of the corresponding variables are concluded by the interpolated postprocessing technique. Numerical tests are provided to justify the performance of the theoretical analysis.