Strong convergence of a fully discrete finite element approximation of the stochastic Cahn-Hilliard equation

Abstract

We consider the stochastic Cahn-Hilliard equation driven by additive Gaussian noise in a convex domain with polygonal boundary in dimension $d\le 3$. We discretize the equation using a standard finite element method in space and a fully implicit backward Euler method in time. By proving optimal error estimates on subsets of the probability space with arbitrarily large probability and uniform-in-time moment bounds we show that the numerical solution converges strongly to the solution as the discretization parameters tend to zero.

arXiv comment25 pages

References in corpus (5)

• Galerkin approximations for the stochastic Burgers equation
• On discretization schemes for stochastic evolution equations
• Finite Element Approximation of the Cahn-Hilliard-Cook equation
• On the backward Euler approximation of the stochastic Allen-Cahn equation
• Finite element approximation of the linearized Cahn-Hilliard-Cook equation

Cited by in corpus (14)

• Optimal error estimates of Galerkin finite element methods for stochastic Allen-Cahn equation with additive noise
• Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations
• Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations
• Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities
• Density convergence of a fully discrete finite difference method for stochastic Cahn--Hilliard equation
• Convergence analysis of a finite difference method for stochastic Cahn--Hilliard equation
• Numerical analysis of a full discretization for stochastic Cahn--Hilliard equation driven by additive noise
• Error estimates of semi-discrete and fully discrete finite element methods for the Cahn-Hilliard-Cook equation
• Strong convergence rates of an explicit scheme for stochastic Cahn--Hilliard equation with additive noise
• Weak convergence of the backward Euler method for stochastic Cahn--Hilliard equation with additive noise
• Asymptotics of large deviations of finite difference method for stochastic Cahn--Hilliard equation
• Absolute continuity and numerical approximation of stochastic Cahn--Hilliard equation with unbounded noise diffusion
• Convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations on 2D torus
• Fully Discrete Mixed Finite Element Methods for the Stochastic Cahn-Hilliard Equation with Gradient-type Multiplicative Noise