Numerical Approximation of the Space Fractional Cahn-Hilliard Equation

Analysis and Approximation of a Fractional Cahn-Hilliard Equation

We derive a Fractional Cahn-Hilliard Equation (FCHE) by considering a gradient flow in the negative order Sobolev space H−α, α ∈ [0, 1] where the choice α = 1 corresponds to the classical Cahn-Hilliard equation whilst the choice α = 0 recovers the Allen-Cahn equation. The existence of a unique solution is established and it is shown that the equation preserves mass for all positive values of fr...

The Cahn-hilliard Equation

1. Steady states. There are many two component systems in which phase separation can be induced by rapidly cooling the system. Thus, if a two component system, which is spatially uniform at temperature T1, is rapidly cooled to a second sufficiently lower temperature T2, then the cooled system will separate into regions of higher and lower concentration. A phenomenological description of the beh...

The Cahn–Hilliard Equation

Finite Element Approximation of the Cahn-Hilliard-Cook Equation

We study the nonlinear stochastic Cahn-Hilliard equation driven by additive colored noise. We show almost sure existence and regularity of solutions. We introduce spatial approximation by a standard finite element method and prove error estimates of optimal order on sets of probability arbitrarily close to 1. We also prove strong convergence without known rate.

Finite Element Approximation of the Linearized Cahn-hilliard-cook Equation

The linearized Cahn-Hilliard-Cook equation is discretized in the spatial variables by a standard finite element method. Strong convergence estimates are proved under suitable assumptions on the covariance operator of the Wiener process, which is driving the equation. The backward Euler time stepping is also studied. The analysis is set in a framework based on analytic semigroups. The main part ...