AbstractWe consider the stochastic Cahn–Hilliard equation with additive noise term $$\varepsilon ^\gamma g\, {\dot{W}}$$
ε
γ
g
W
˙
($$\gamma >0$$
γ
>
0
) that scales with the interfacial width parameter $$\varepsilon $$
ε
. We verify strong error estimates for a gradient flow structure-inheriting time-implicit discretization, where $$\varepsilon ^{-1}$$
ε
-
1
only enters polynomially; the proof is based on higher-moment estimates for iterates, and a (discrete) spectral estimate for its deterministic counterpart. For $$\gamma $$
γ
sufficiently large, convergence in probability of iterates towards the deterministic Hele–Shaw/Mullins–Sekerka problem in the sharp-interface limit $$\varepsilon \rightarrow 0$$
ε
→
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’ $$\gamma $$
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 $$\gamma \ge 1$$
γ
≥
1
is the deterministic problem, and for $$\gamma =0$$
γ
=
0
we obtain agreement with a (new) stochastic version of the Mullins–Sekerka problem.
Strong Convergence of a Fully Discrete Finite Element Approximation of the Stochastic Cahn--Hilliard Equation
