Cahn-Hilliard方程多重网格求解器收敛性分析

Convergence Analysis of Multigrid Solver for Cahn-Hilliard Equation

Cahn-Hilliard(CH)方程是相场模型中的一个基本的非线性方程,通常使用数值方法进行分析。在对CH方程进行数值离散后会得到一个非线性的方程组,全逼近格式(Full Approximation Storage,FAS)是求解这类非线性方程组的一个高效多重网格迭代格式。目前众多的求解CH方程主要关注数值格式的收敛性,而没有论证求解器的可靠性。文中给出了求解CH方程离散得到的非线性方程组的多重网格算法的收敛性证明,从理论上保证了计算过程的可靠性。针对CH方程的时间二阶全离散差分数值格式,利用快速子空间下降(Fast Subspace Descent,FASD)框架给出其FAS格式多重网格求解器的收敛常数估计。为了完成这一目标,首先将原本的差分问题转化为完全等价的有限元问题,再论证有限元问题来自一个凸泛函能量形式的极小化,然后验证能量形式及空间分解满足FASD框架假设,最终得到原多重网格算法的收敛系数估计。结果显示,在非线性情形下,CH方程中的参数ε对网格尺度添加了限制,太小的参数会导致数值计算过程不收敛。最后通过数值实验验证了收敛系数与方程参数及网格尺度的依赖关系。

The Cahn-Hilliard(CH)equation is a fundamental nonlinear equation in the phase field model and is usually analyzed using numerical methods.Following a numerical discretization,we get a nonlinear equations system.The full approximation scheme(FAS)is an efficient multigrid iterative scheme for solving such nonlinear equations.In the numerous articles on solving the CH equation,the main focus is on the convergence of the numerical format,without mentioning the stability of the solver.In this paper,the convergence property of the multigrid algorithm is established,which is from the nonlinear equation system obtained by solving the discrete CH equation,and the reliability of the calculation process is guaranteed theoretically.For the diffe-rence discrete numerical scheme of the CH equation,which is both second-order in spatial and time,we use the fast subspace descent method(FASD)framework to give the estimation of the convergence constant of its FAS scheme multigrid solver.First,we transform the original difference problem into a fully equivalent finite element problem.It demonstrates that the finite element problem comes from the minimization of convex functional energy.Then it is verified that the energy functional and the spatial decomposition satisfy the FASD framework assumption.Finally,the convergence coefficient estimate of the original multigrid algorithm is obtained.The results show that in the case of nonlinearity,the parameterεin the CH equation imposes restrictions on the grid size,which will cause the numerical calculation process not to converge when it is too small.Finally,the spatial and temporal accuracy of the numerical format is verified by numerical experiment,and the dependence of the convergence coefficient on the equation parameters and grid-scale is analyzed.