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

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.