Continuous data assimilation and long-time accuracy in a C<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.svg"><mml:msup><mml:mrow /><mml:mn>0</mml:mn></mml:msup></mml:math> interior penalty method for the Cahn-Hilliard equation

On the Stationary Cahn-hilliard Equation: Interior Spike Solutions

We study solutions of the stationary Cahn-Hilliard equation in a bounded smooth domain which have a spike in the interior. We show that a large class of interior points (the “nondegenerate peak” points) have the following property: there exist such solutions whose spike lies close to a given nondegenerate peak point. Our construction uses among others the methods of viscosity solution, weak con...

A discontinuous Galerkin method for the Cahn-Hilliard equation

A discontinuous Galerkin finite element method has been developed to treat the high-order spatial derivatives appearing in the Cahn–Hilliard equation. The Cahn–Hilliard equation is a fourth-order nonlinear parabolic partial differential equation, originally proposed to model phase segregation of binary alloys. The developed discontinuous Galerkin approach avoids the need for mixed finite elemen...

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

Analysis of Mixed Interior Penalty Discontinuous Galerkin Methods for the Cahn-Hilliard Equation and the Hele-Shaw Flow

This paper proposes and analyzes two fully discrete mixed interior penalty discontinuous Galerkin (DG) methods for the fourth order nonlinear Cahn-Hilliard equation. Both methods use the backward Euler method for time discretization and interior penalty discontinuous Galerkin methods for spatial discretization. They differ from each other on how the nonlinear term is treated, one of them is bas...