Abstract:

The Cahn-Hilliard equation is an important class of fourth-order nonlinear diffusion equations with rich physical background and profound research value. In the numerical simulation, the existence of the nonlinear potential term $f(u)$ of the equation and the strong rigidity caused by small parameter $\epsilon$ will bring many challenges, so it is important to design efficient and accurate numerical schemes to satisfy the discrete energy law of the equations. In this paper, the Cahn-Hilliard equation is solved by the discontinuous finite volume element method (DFVEM) combined with the fully implicit scheme, and the important theoretical results of mass conservation and energy dissipation in the fully discrete scheme are proved. At the same time, the semi-discrete format error estimates are given. Finally, numerical experiments propose an adaptive time stepping strategy and verify the effectiveness of the method.

Key words: The Cahn-Hilliard equation, Discontinuous finite volume element method, Discrete energy dissipation