关键词: 非线性多重网格, 收敛性分析, Cahn-Hilliard方程, 全逼近格式, 非线性问题

Abstract: 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 parameter ε in 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.

Key words: Nonlinear multigrid, Convergence analysis, Cahn-Hilliard equation, Full approximation scheme, Nonlinear problem