Speaker
Description
The Cahn-Hilliard equation is a deterministic model for the description of phase separation processes in metal alloys, which occur if the alloy is rapidly cooled below a critical temperature. This equation can be interpreted as an $H^{-1}$-gradient flow of the Ginzburg-Landau energy functional, which consists of a gradient term and double-well potential favoring phase separation. If the quench is shallow, i.e. the temperature is still close to the critical temperature, the double-well potential can be approximated by a smooth fourth-order polynomial. Yet, in the deep quench limit, i.e. when the temperature is significantly smaller than the critical temperature, a singular double-obstacle potential is the better choice. It is also well-known that in particular the early stages of the separation process are heavily influenced by thermal fluctuations which are not included in the deterministic description.
In this talk, we discuss the numerical treatment of the stochastic Cahn-Hilliard equation with double-obstacle potential and conservative noise on a periodic domain. In particular, we propose a fully discrete finite element scheme and present a convergence result. In this endeavor, special attention has to be paid to the interplay between the singularities of the potential and the stochastic forcing term. In comparison to the stochastic Allen-Cahn equation, which is based on an $L^2$-gradient flow, balancing these challenges in the case of the Cahn-Hilliard equation poses additional difficulties due to the underlying $H^{-1}$-structure. Conceptually, our proof relies on monotonicity arguments and omits the application of Skorokhod's theorem, which allows us to show convergence towards probabilistically strong solutions.
We conclude by presenting numerical simulations underlining the practicality of the proposed scheme and the importance of the additional stochastic fluxes.