Analysis of a mixed finite element method for stochastic Cahn-Hilliard equation with multiplicative noise

TL;DR

This paper introduces a new fully discrete finite element scheme for stochastic Cahn-Hilliard equations with multiplicative noise, proving stability, higher moment stability, and strong convergence, supported by numerical experiments.

Contribution

The paper presents a novel finite element scheme with stability and convergence analysis for stochastic Cahn-Hilliard equations with multiplicative noise, addressing challenges from nonlinearity and noise interaction.

Findings

Proved $L^2$-stability and discrete $H^2$-stability of the scheme.

Established higher moment stability in $L^2$-norm.

Demonstrated strong convergence and validated results through numerical experiments.

Abstract

This paper proposes and analyzes a novel fully discrete finite element scheme with the interpolation operator for stochastic Cahn-Hilliard equations with functional-type noise. The nonlinear term satisfies a one-side Lipschitz condition and the diffusion term is globally Lipschitz continuous. The novelties of this paper are threefold. First, the $L^2$-stability ($L^\infty$ in time) and the discrete $H^2$-stability ($L^2$ in time) are proved for the proposed scheme. The idea is to utilize the special structure of the matrix assembled by the nonlinear term. None of these stability results has been proved for the fully implicit scheme in existing literature due to the difficulty arising from the interaction of the nonlinearity and the multiplicative noise. Second, the higher moment stability in $L^2$-norm of the discrete solution is established based on the previous stability results. Third, the H\"older continuity in time for the strong solution is established under the minimum assumption of the strong solution. Based on these, the discrete $H^{-1}$-norm of the strong convergence is discussed. Several numerical experiments including stability and convergence are also presented to validate our theoretical results.