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# Communications in Mathematical Sciences

## Volume 14 (2016)

### Number 2

### An $H^2$ convergence of a second-order convex-splitting, finite difference scheme for the three-dimensional Cahn–Hilliard equation

Pages: 489 – 515

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n2.a8

#### Authors

#### Abstract

In this paper we present an unconditionally solvable and energy stable second order numerical scheme for the three-dimensional (3D) Cahn–Hilliard (CH) equation. The scheme is a twostep method based on a second order convex splitting of the physical energy, combined with a centered difference in space. The equation at the implicit time level is nonlinear but represents the gradients of a strictly convex function and is thus uniquely solvable, regardless of time step-size. The nonlinear equation is solved using an efficient nonlinear multigrid method. In addition, a global in time $H^2_h$ bound for the numerical solution is derived at the discrete level, and this bound is independent on the final time. As a consequence, an unconditional convergence (for the time step $s$ in terms of the spatial grid size $h$) is established, in a discrete $L^{\infty}_s (0, T; H^2_h$) norm, for the proposed second order scheme. The results of numerical experiments are presented and confirm the efficiency and accuracy of the scheme.

#### Keywords

Cahn–Hilliard equation, finite difference, second-order, energy stability, multigrid, global-in-time $H^2_h$ stability, $L^{\infty}_s (0, T; H^2)$ convergence analysis

#### 2010 Mathematics Subject Classification

35K30, 65M12, 65M55