大密度比相场模型的高效有限元方法

The phase field model with large density ratio describes well the evolution of two-phase flow with large differences in density, which has an extremely important application in industry and economy. Therefore, it is very important to simulate the model effectively. Compared to the two-phase flow with the same or similar densities, the phase field model with large density ratio can be applied more widely, but it also brings greater challenges to the design and analysis of the algorithm.This project is expected to make some innovative work in numerical simulation and theoretical analysis of phase-field models with large density ratio. The phase field model is a combination of variable-density Navier-Stokes equations and Cahn-Hilliard equations. First, the error estimates for the finite element discretization will be respectively analyzed for some efficient numerical schemes of these two equations. For Cahn-Hilliard equations, a linear finite element method based on the gradient recovery will be constructed. It will be proved that the algorithm has the same optimal or superconvergence accuracy as it solves the biharmonic equation numerically. Then based on the above results, we will construct a linear, decoupled and unconditionally stable finite element method for the phase field model and analyze the corresponding error estimates. Finally, a fully multigrid algorithm for nonlinear parabolic equations will be constructed, which can be proved to achieve the accuracy on the finest grid but reduce the computation cost significantly. The fully multigrid algorithm will also be applied to the phase field model.

大密度比相场模型很好地描述了密度差别较大的两相流混合的演变过程，在工业和经济中有着极其重要的应用，因此有效地数值模拟该模型起着至关重要的作用。相比密度相同或相近的两相流，大密度比相场模型适应范围更广，但是给算法的设计和分析也带来更大的挑战。本项目预期在大密度比相场模型的数值模拟和理论分析方面做出一些创新性工作。该相场模型是由变密度Navier-Stokes方程和Cahn-Hilliard方程耦合而成。本项目首先分别对这两方程的某些高效数值格式分析有限元离散的误差估计，针对Cahn-Hilliard方程还构造基于梯度重构的线性有限元算法，证明与其计算双调和方程一样具有最优阶或超收敛精度。然后在此基础上构造大密度比相场模型解耦、无条件稳定的线性格式并分析误差。最后，构造非线性抛物方程的完全多重网格算法，证明其保证最细网格精度的同时大大节省计算量，并应用到上述相场模型。

大密度比相场模型很好地描述了密度差别较大的两相流混合的演变过程，在工业和经济中有着极其重要的应用，因此有效地数值模拟该模型起着至关重要的作用。相比密度相同或相近的两相流，大密度比相场模型适应范围更广，但是给算法的设计和分析也带来更大的挑战。本项目在大密度比相场模型的数值模拟和理论分析方面做出了一些创新性工作，包括针对变密度Navier-Stokes方程构造了高效解耦的Gauge-Uzawa算法以及基于标量辅助变量的自适应算法，针对Cahn-Hilliard方程构造了基于标量辅助变量的有限元算法，针对蒙日安培方程构造了基于梯度重构的线性有限元算法，以及针对推广的 Kawahara 方程、Boussinesq Paradigm 方程和耦合 Schrodinger-Boussinesq 方程也都构造了基于标量辅助变量的高效算法。针对上述算法，本项目不仅从理论上证明了上述解耦格式的高效性、无条件稳定性，物理量保正或守恒，还推导了相应的误差估计。最后，数值实验也验证了理论结果，证实了上述算法不仅具有最优阶的计算精度，也大大节省了计算时间。

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