Abstract
In this paper, we employ the multilevel Monte Carlo finite element method to solve the stochastic Cahn–Hilliard–Cook equation. The Ciarlet–Raviart mixed finite element method is applied to solve the fourth-order equation. In order to estimate the mild solution, we use finite elements for space discretization and the semi-implicit Euler–Maruyama method in time. For the stochastic scheme, we use the multilevel method to decrease the computational cost (compared to the Monte Carlo method). We implement the method to solve three specific numerical examples (both two- and three dimensional) and study the effect of different noise measures.
Highlights
The Cahn–Hilliard equation is a robust mathematical model for describing different phase separation phenomena, from co-polymer systems to lipid membranes
For the time-dependent stochastic problems, the total error consists of the spatial error, the time discretization error and the statistical error
Using the obtained results, we achieve the error bound of the multilevel Monte Carlo considering the principal discretization error, i.e., spatial discretization, time stepping errors and statistical error
Summary
The Cahn–Hilliard equation is a robust mathematical model for describing different phase separation phenomena, from co-polymer systems to lipid membranes. For the time-dependent stochastic problems, the total error consists of the spatial error (due to the finite element method), the time discretization error (due to the Euler–Maruyama technique) and the statistical error (number of samples). The multilevel Monte Carlo method uses hierarchies of meshes for time and space approximations in the sense that the number of samples and mesh sizes (as well as time steps) on the different levels are chosen such that the errors are equilibrated. For the stochastic Cahn–Hilliard–Cook equation, we strive to determine an optimal hierarchy of meshes, number of samples and time intervals which minimize the total computational work. We use the MLMC-FEM for the fourth-order stochastic equations and calculate the mild solution of the Cahn–Hilliard–Cook equation. We strive to minimize the computational complexity with respect to a given error tolerance This procedure is compared to the Monte Carlo method.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
