A Positivity-Preserving Finite Element Scheme for the Relaxed Cahn-Hilliard Equation with Single-Well Potential and Degenerate Mobility
Résumé
We propose and analyse a finite element approximation of the Cahn-Hilliard equation regularised in space with single-well potential of Lennard-Jones type and degenerate mobility. 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 in one and two dimensions and we show that it preserves the physical bounds of the solutions thanks to an upwind approach adapted to the finite elements method. Moreover, we show well-posedness, energy stability properties and convergence of solutions to the numerical scheme. Finally, numerical simulations in one and two dimensions are presented.
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