2019/03/12 by Xiaobing Feng, Yukun Li, Feng, Xiaobing +3
Computer Science · Engineering · Materials Science · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Solidification and crystal growth phenomena
paper · pdf · doi:10.48550/arxiv.1903.05146
openalex publication_date 2019/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper develops and analyzes some fully discrete mixed finite element methods for the stochastic Cahn-Hilliard equation with gradient-type multiplicative noise that is white in time and correlated in space. The stochastic Cahn-Hilliard equation is formally derived as a phase field formulation of the stochastically perturbed Hele-Shaw flow. The main result of this paper is to prove strong convergence with optimal rates for the proposed mixed finite element methods. To overcome the difficulty caused by the low regularity in time of the solution to the stochastic Cahn-Hilliard equation, the Hölder continuity in time with respect to various norms for the stochastic PDE solution is established, and it plays a crucial role in the error analysis. Numerical experiments are also provided to validate the theoretical results and to study the impact of noise on the Hele-Shaw flow as well as the interplay of the geometric evolution and gradient-type noise.