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Numerical approximation of the Stochastic Cahn-Hilliard Equation near the Sharp Interface Limit
Antonopoulou, Dimitra Banas, Lubomir Nurnberg, Robert Prohl, Andreas
Antonopoulou, Dimitra
Banas, Lubomir
Nurnberg, Robert
Prohl, Andreas
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Abstract
Abstract. We consider the stochastic Cahn-Hilliard equation with additive noise term
that scales with the interfacial width parameter ε. We verify strong error
estimates for a gradient flow structure-inheriting time-implicit discretization, where ε
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 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’ 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 ≥ 1 is the deterministic problem,
and for γ = 0 we obtain agreement with a (new) stochastic version of the Mullins-Sekerka
problem.
Citation
Antonopoulou, D., Banas, L., Nurnberg, R. & Prohl, A. (2021). Numerical approximation of the Stochastic Cahn-Hilliard Equation near the Sharp Interface Limit. Numerische Mathematik, 147, 505-551. https://doi.org/10.1007/s00211-021-01179-7
Publisher
Springer
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Numerische Mathematik
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DOI
10.1007/s00211-021-01179-7
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0029-599X
EISSN
0945-3245