Numerical approximation of the stochastic Cahn\u2013Hilliard equation near the sharp interface limit

Abstract

We consider the stochastic Cahn–Hilliard equation with additive noise term varepsilon ^gamma g, {dot{W}} (gamma >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 gamma 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’ gamma ) 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 gamma ge 1 is the deterministic problem, and for gamma =0 we obtain agreement with a (new) stochastic version of the Mullins–Sekerka problem.