A fully decoupled numerical method for Cahn–Hilliard–Navier–Stokes–Darcy equations based on auxiliary variable approaches

A fully decoupled, linearized, and unconditionally stable finite element method is developed to solve the Cahn-Hilliard-Navier–Stokes-Darcy model in coupled free fluid region porous medium region. By introducing two auxiliary energy variables, we derive equivalent system that consistent with original system. The dissipation law of proposed proven. To lay a solid foundation, first present linearized time-stepping for reformulated system, prove its stability. In order further improve computational efficiency, special treatment interface conditions artificial compression approach are utilized decouple subdomains Navier–Stokes equation. Therefore, discretization techniques existing variable approaches, decoupled numerical scheme can be developed, under framework semi-implicit semi-explicit temporal discretizaion Galerkin spatial discretization. grad-div stabilization also employed stability algorithm. full obeys desired without any restriction. Moreover, implementation process discussed, including adaptive mesh strategy accurately capture diffuse interface. Ample experiments performed validate typical features schemes, such as accuracy, restriction time step size, refinement space. Furthermore, apply simulate shape relaxation Buoyancy-driven flows, which demonstrate applicability method.