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This paper develops and analyzes some fully discrete mixed finite elementmethods for the stochastic Cahn-Hilliard equation with gradient-typemultiplicative noise that is white in time and correlated in space. Thestochastic Cahn-Hilliard equation is formally derived as a phase fieldformulation of the stochastically perturbed Hele-Shaw flow. The main result ofthis paper is to prove strong convergence with optimal rates for the proposedmixed finite element methods. To overcome the difficulty caused by the lowregularity in time of the solution to the stochastic Cahn-Hilliard equation,the H\"{o}lder continuity in time with respect to various norms for thestochastic PDE solution is established, and it plays a crucial role in theerror analysis. Numerical experiments are also provided to validate thetheoretical results and to study the impact of noise on the Hele-Shaw flow aswell as the interplay of the geometric evolution and gradient-type noise.

Fully Discrete Mixed Finite Element Methods for the Stochastic Cahn-Hilliard Equation with Gradient-type Multiplicative Noise