Quantum mechanics

Applied mathematics

Economics

Thermodynamics

Physics

Finite element method

Nonlinear system

Mathematics

Mathematical analysis

Financial economics

Numerical analysis

Convexity

Partial differential equation

Cahn–Hilliard equation

Energy functional

Journal of Scientific Computing

In this article, we present and analyze a finite element numerical scheme for a three-component macromolecular microsphere composite (MMC) hydrogel model, which takes the form of a ternary Cahn–Hilliard-type equation with Flory–Huggins–deGennes energy potential. The numerical approach is based on a convex–concave decomposition of the energy functional in multi-phase space, in which the logarithmic and the nonlinear surface diffusion terms are treated implicitly, while the concave expansive linear terms are explicitly updated. A mass lumped finite element spatial approximation is applied, to ensure the positivity of the phase variables. In turn, a positivity-preserving property can be theoretically justified for the proposed fully discrete numerical scheme. In addition, unconditional energy stability is established as well, which comes from the convexity analysis. Several numerical simulations are carried out to verify the accuracy and positivity-preserving property of the proposed scheme.

An Energy Stable Finite Element Scheme for the Three-Component Cahn–Hilliard-Type Model for Macromolecular Microsphere Composite Hydrogels