In this paper, we propose a discontinuous finite volume element method to solve a phase field model for two immiscible incompressible fluids. In this finite volume element scheme, discontinuous linear finite element basis functions are used to approximate the velocity, phase function, and chemical potential while piecewise constants are used to approximate the pressure. This numerical method is efficient, optimally convergent, conserving the mass, convenient to implement, flexible for mesh refinement, and easy to handle complex geometries with different types of boundary conditions. We rigorously prove the mass conservation property and the discrete energy dissipation for the proposed fully discrete discontinuous finite volume element scheme. Using numerical tests, we verify the accuracy, confirm the mass conservation and the energy law, test the influence of surface tension and small density variations, and simulate the driven cavity, the Rayleigh-Taylor instability.

中文翻译：

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Navier-Stokes-Cahn-Hilliard耦合相场模型的间断有限体积元方法

在本文中，我们提出了一种不连续的有限体积元方法来求解两种不可混溶不可压缩流体的相场模型。在此有限体积元方案中，不连续的线性有限元基函数用于近似速度，相函数和化学势，而分段常数用于近似压力。该数值方法高效，最优收敛，节省质量，易于实现，网格细化灵活并且易于处理具有不同类型边界条件的复杂几何形状。对于提出的完全离散的不连续有限体积单元方案，我们严格证明了质量守恒性质和离散的能量耗散。通过数值测试，我们验证了准确性，确认了质量守恒和能量定律，