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

Lixiu Dong,Cheng Wang,Hui Zhang,Zhengru Zhang

Published 2019 in Communications in Mathematical Sciences

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 `∞(0,T ;H−1 h )∩` 2(0,T ;H1 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.

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