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# Communications in Mathematical Sciences

## Volume 17 (2019)

### Number 4

### A positivity-preserving, energy stable and convergent numerical scheme for the Cahn–Hilliard equation with a Flory–Huggins–Degennes energy

Pages: 921 – 939

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n4.a3

#### Authors

#### Abstract

This article is focused on the bound estimate and convergence analysis of an unconditionally energy-stable scheme for the MMC-TDGL equation, a Cahn–Hilliard equation with a Flory–Huggins–deGennes energy. The numerical scheme, a finite difference algorithm based on a convex splitting technique of the energy functional, was proposed in [*Sci. China Math.*, 59:1815, 2016]. We provide a theoretical justification of the unique solvability for the proposed numerical scheme, in which a well-known difficulty associated with the singular nature of the logarithmic energy potential has to be handled. Meanwhile, a careful analysis reveals that, such a singular nature prevents the numerical solution of the phase variable reaching the limit singular values, so that the positivitypreserving property could be proved at a theoretical level. In particular, the natural structure of the deGennes diffusive coefficient also ensures the desired positivity-preserving property. In turn, the unconditional energy stability becomes an outcome of the unique solvability and the convex-concave decomposition for the energy functional. Moreover, an optimal rate convergence analysis is presented in the $\ell^\infty (0, T; H^{-1}_h) \cap \ell^2 (0, T; H^1_h)$ norm, in which the the convexity of nonlinear energy potential has played an essential role. In addition, a rewritten form of the surface diffusion term has facilitated the convergence analysis, in which we have made use of the special structure of concentration-dependent deGennes type coefficient. Some numerical results are presented as well.

#### Keywords

Cahn–Hilliard equation, Flory–Huggins energy, deGennes diffusive coefficient, energy stability, positivity-preserving, convergence analysis

#### 2010 Mathematics Subject Classification

60F10, 60J75, 62P10, 92C37

This work is supported in part by the grants NSF DMS-1418689 (C. Wang), NSFC-11471046 and 11571045 (H. Zhang), NSFC-11571045, the Science Challenge Project TZ2018002 and the Fundamental Research Funds for the Central Universities (Z. Zhang).

Received 10 June 2018

Accepted 15 February 2019

Published 25 October 2019