This paper is devoted to the numerical approximations of the Cahn-Hilliard-Darcy-Stokes system, which is a combination of the modified Cahn-Hilliard equation with the Darcy-Stokes equation. A novel discontinuous Galerkin pressure-correction scheme is proposed for solving the coupled system, which can achieve the desired level of linear, fully decoupled, and unconditionally energy stable. The developed scheme here is implemented by combining several effective techniques, including by adding an additional stabilization term artificially in Cahn-Hilliard equation for balancing the explicit treatment of the coupling term, the stabilizing strategy for the nonlinear energy potential, and a rotational pressure-correction scheme for the Darcy-Stokes equation. We rigorously prove the unique solvability, unconditional energy stability, and optimal error estimates of the proposed scheme. Finally, a number of numerical examples are provided to demonstrate numerically the efficiency of the present formulation.

本文致力于 Cahn-Hilliard-Darcy-Stokes 系统的数值近似，该系统是修正 Cahn-Hilliard 方程与 Darcy-Stokes 方程的组合。提出了一种新颖的间断伽辽金压力修正方案来求解耦合系统，该方案可以达到所需的线性、完全解耦和无条件能量稳定的水平。这里开发的方案是通过结合几种有效的技术来实现的，包括在 Cahn-Hilliard 方程中人为地添加一个额外的稳定项来平衡耦合项的显式处理、非线性能势的稳定策略以及旋转压力校正达西-斯托克斯方程的方案。我们严格证明了所提出方案的独特可解性、无条件能量稳定性和最优误差估计。最后，提供了许多数值例子来从数值上证明本公式的效率。