Written originally to model phase separation in binary alloys, the Cahn-Hilliard Equation has also been applied to problems as diverse as multi-phase fluid flow, image processing and planet formation. It is a parabolic partial differential with an instability-promoting second-order spatial derivative, but is stabilized by a fourth-order derivative. The competition between these terms leads to the formation of diffuse interfaces with tunable thickness. This equation has also spawned a burgeoning mathematical literature. In this talk I will introduce this equation from a physical standpoint, and proceed to a discontinuous Galerkin
(DG) finite element method designed to treat the high-order derivatives. The crux of the talk is that $C^0$ interpolation functions are sufficient in the DG setting despite the high-order derivatives. The weak form of the problem is the point of departure for proving consistency of the method, boundedness of the time-continuous solutions, and stability of the time-discrete solutions.
Several
numerical examples and a convergence study will be presented.
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