Numerical approximation of the stochastic Cahn–Hilliard equation near the sharp interface limit

• Published in: Numerische Mathematik 2021-03, Vol.147 (3), p.505-551
• Main Authors: Antonopoulou, Dimitra, Baňas, Ĺubomír, Nürnberg, Robert, Prohl, Andreas
• Format: Article
• Language: English
• Subjects: Convergence; Discretization; Gradient flow; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical Analysis; Numerical and Computational Physics; Parameter estimation; Simulation; Theoretical
• ISSN: 0029-599X; 0945-3245
• DOI: 10.1007/s00211-021-01179-7

We consider the stochastic Cahn–Hilliard equation with additive noise term ε γ g W ˙ ( γ > 0 ) that scales with the interfacial width parameter ε . We verify strong error estimates for a gradient flow structure-inheriting time-implicit discretization, where 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 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.