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多级蒙特卡洛有限元方法求解对数势Cahn-Hilliard-Cook方程误差分析
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Abstract:
本文用多级蒙特卡洛有限元方法求解具有对数势的随机Cahn-Hilliard-Cook方程。为了估计方程的温和解，运用Ciarlet-Raviart有限元方法进行空间离散化，对时间则采用向后欧拉格式离散，得到方程的全离散数值格式。同时运用多级蒙特卡洛方法进行数值模拟，相较于标准蒙特卡洛方法，提高了计算效率。文中主要给出了全离散格式的误差估计以及分别应用标准蒙特卡洛方法和多级蒙特卡洛方法进行数值模拟时的总误差估计。
In this paper, a multilevel Monte Carlo finite element method is used to solve the stochastic Cahn-Hilliard-Cook equation with logarithmic potential. To estimate the mild solution of the equation, the Ciarlet-Raviart finite element method is applied for spatial discretization and the backward Euler scheme is applied for time discretization to obtain the full discrete numerical scheme of the equation. Numerical simulations were conducted using the multilevel Monte Carlo method, which enhances computational efficiency compared to the standard Monte Carlo method. The error estimates for the fully discrete scheme and the total error estimates are provided for the numerical simulations using the standard Monte Carlo method and the multilevel Monte Carlo method, respectively.
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