We propose and analyze a structure-preserving space-time variational discretization method for the Cahn-Hilliard-Navier-Stokes system. Uniqueness and stability for the discrete problem is established in the presence of concentration dependent mobility and viscosity parameters by means of the relative energy estimates and order optimal convergence rates are established for all variables using balanced approximation spaces and relaxed regularity conditions on the solution. Numerical tests are presented to demonstrate the proposed method is fully practical and yields the predicted convergence rates. The discrete stability estimates developed in this paper may also be used to analyse other discretization schemes, which is briefly outlined in the discussion.

翻译：本文针对Cahn-Hilliard-Navier-Stokes系统提出并分析了一种保结构时空变分离散方法。通过相对能量估计，在存在浓度依赖的迁移率和粘度参数的情况下，建立了离散问题的唯一性与稳定性；并利用平衡逼近空间和放宽的解正则性条件，给出了所有变量的最优阶收敛率。数值实验表明，所提方法完全实用，并能够实现预期的收敛速率。本文所发展的离散稳定性估计也可用于分析其他离散格式，这将在讨论部分中简要概述。