Continuous data assimilation and long-time accuracy in a C0 interior penalty method for the Cahn-Hilliard equation

• Published in: Applied mathematics and computation 2022-07, Vol.424, Article 127042
• Main Authors: Diegel, Amanda E., Rebholz, Leo G.
• Format: Article
• Language: English
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2022.127042

•Applies continuous data assimilation (CDA) to Cahn-Hilliard.•Prove long time accuracy given enough measurement data.•We use 4th order Cahn-Hilliard with C0 interior penalty•Excellent numerical results for shape relaxation and dumbbell region. We propose a numerical approximation method for the Cahn-Hilliard equations that incorporates continuous data assimilation in order to achieve long time accuracy. The method uses a C0 interior penalty spatial discretization of the fourth order Cahn-Hilliard equations, together with a backward Euler temporal discretization. We prove the method is long time stable and long time accurate, for arbitrarily inaccurate initial conditions, provided enough data measurements are incorporated into the simulation. Numerical experiments illustrate the effectiveness of the method on a benchmark test problem.