A Fully Discrete Mixed Finite Element Method for the Stochastic Cahn–Hilliard Equation with Gradient-Type Multiplicative Noise

Abstract

This paper develops and analyzes a fully discrete mixed finite element method for the stochastic Cahn–Hilliard equation with gradient-type multiplicative noise that is white in time and correlated in space. The stochastic Cahn–Hilliard equation is formally derived as a phase field formulation of the stochastically perturbed Hele–Shaw flow. The main result of this paper is to prove strong convergence with optimal rates for the proposed mixed finite element method. To overcome the difficulty caused by the low regularity in time of the solution to the stochastic Cahn–Hilliard equation, the Holder continuity in time with respect to various norms for the stochastic PDE solution is established, and it plays a crucial role in the error analysis. Numerical experiments are also provided to validate the theoretical results and to study the impact of noise on the Hele–Shaw flow as well as the interplay of the geometric evolution and gradient-type noise.