Divergence-free and mass-conservative virtual element methods for the Navier-Stokes-Cahn-Hilliard system

TL;DR

This paper develops and analyzes divergence-free, mass-conservative virtual element methods for simulating two-phase fluid flows governed by the Navier-Stokes-Cahn-Hilliard system, ensuring stability and accuracy.

Contribution

It introduces a novel variational formulation and high-order divergence-free virtual element discretizations that preserve mass and energy, with proven error estimates and numerical validation.

Findings

Methods satisfy mass conservation and energy bounds.

Optimal error estimates are established.

Numerical experiments confirm theoretical results and good performance.

Abstract

In this work, we design and analyze semi/fully-discrete virtual element approximations for the time-dependent Navier--Stokes-Cahn--Hilliard equations, modeling the dynamics of two-phase incompressible fluid flows with diffuse interfaces. A new variational formulation is derived involving solely the velocity, pressure, and phase field, together with corresponding a priori energy estimates. The spatial discretization is based on the coupling divergence-free and $C^1$-conforming elements of high-order, while the time discretization employs a classical backward Euler scheme. By introducing a novel skew-symmetric trilinear form to discretize the convective term in the Cahn--Hilliard equation, we propose discrete schemes that satisfy mass conservation and energy bounds. Moreover, optimal error estimates are provided for both formulations. Finally, two numerical experiments are presented to support our theoretical findings and to illustrate the good performance of the proposed schemes for different polynomial degrees and polygonal meshes.