A nonnegativity preserving scheme for the relaxed Cahn-Hilliard equation with single-well potential and degenerate mobility

TL;DR

This paper introduces a finite element scheme for the relaxed Cahn-Hilliard equation with singular potential and degenerate mobility, ensuring energy stability and nonnegativity, validated through numerical simulations.

Contribution

It presents a novel finite element method that preserves physical bounds and guarantees energy stability for a complex Cahn-Hilliard model with biological relevance.

Findings

Scheme preserves nonnegativity and physical bounds.

Proves well-posedness and energy stability.

Numerical simulations confirm effectiveness.

Abstract

We propose and analyze a finite element approximation of the relaxed Cahn-Hilliard equation with singular single-well potential of Lennard-Jones type and degenerate mobility that is energy stable and nonnegativity preserving. The Cahn-Hilliard model has recently been applied to model evolution and growth for living tissues: although the choices of degenerate mobility and singular potential are biologically relevant, they induce difficulties regarding the design of a numerical scheme. We propose a finite element scheme and we show that it preserves the physical bounds of the solutions thanks to an upwind approach adapted to the finite element method. Moreover, we show well-posedness, energy stability properties, and convergence of solutions of the numerical scheme. Finally, we validate our scheme by presenting numerical simulations in one and two dimensions.