Numerical approximation of the Stochastic Cahn-Hilliard Equation near the Sharp Interface Limit

Abstract

We consider the stochastic Cahn-Hilliard equation with additive noise term $\varepsilon^γg\, \dot{W}$ ($γ>0$) that scales with the interfacial width parameter $\varepsilon$. We verify strong error estimates for a gradient flow structure-inheriting time-implicit discretization, where $\varepsilon^{-1}$ only enters polynomially; the proof is based on higher-moment estimates for iterates, and a (discrete) spectral estimate for its deterministic counterpart. For $γ$ sufficiently large, convergence in probability of iterates towards the deterministic Hele-Shaw/Mullins-Sekerka problem in the sharp-interface limit $\varepsilon \rightarrow 0$ is shown. These convergence results are partly generalized to a fully discrete finite element based discretization. We complement the theoretical results by computational studies to provide practical evidence concerning the effect of noise (depending on its 'strength' $γ$) on the geometric evolution in the sharp-interface limit. For this purpose we compare the simulations with those from a fully discrete finite element numerical scheme for the (stochastic) Mullins-Sekerka problem. The computational results indicate that the limit for $γ\geq 1$ is the deterministic problem, and for $γ=0$ we obtain agreement with a (new) stochastic version of the Mullins-Sekerka problem.