On the mass-conserving Allen-Cahn approximation for incompressible binary fluids

Motion of a Droplet for the Stochastic Mass-Conserving Allen-Cahn Equation

We study the stochastic mass-conserving Allen-Cahn equation posed on a smoothly bounded domain of R2 with additive, spatially smooth, space-time noise. This equation describes the stochastic motion of a small almost semicircular droplet attached to domain’s boundary and moving towards a point of locally maximum curvature. We apply Itô calculus to derive the stochastic dynamics of the center of ...

Motion of a Droplet for the Mass-conserving Stochastic Allen-cahn Equation

We study the stochastic mass-conserving Allen-Cahn equation posed on a bounded domain of R2 with additive spatially smooth space-time noise. This equation associated with a small positive parameter ε describes the stochastic motion of a small almost semicircular droplet attached to domain’s boundary and moving towards a point of locally maximum curvature. We apply Itô calculus to derive the sto...

2 Finite Element Approximation of an Allen - Cahn / Cahn - Hilliard

Error control for the approximation of Allen-Cahn and Cahn-Hilliard equations with a logarithmic potential

A fully computable upper bound for the finite element approximation error of Allen– Cahn and Cahn–Hilliard equations with logarithmic potentials is derived. Numerical experiments show that for the sharp interface limit this bound is robust past topological changes. Modifications of the abstract results to derive quasi-optimal error estimates in different norms for lowest order finite element me...

Error estimates for time discretizations of Cahn-Hilliard and Allen-Cahn phase-field models for two-phase incompressible flows

We carry out rigorous error analysis for some energy stable time discretization schemes developed in [20] for a Cahn-Hilliard phase-field model and in [19] for an Allen-Cahn phase-field model.