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arXiv · 1910.13211

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

Alexandre Poulain·Federica Bubba
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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.

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Alexandre Poulain, Federica Bubba. 2022-12-20. A nonnegativity preserving scheme for the relaxed Cahn-Hilliard equation with single-well potential and degenerate mobility. https://doi.org/10.1051/m2an%2F2022050

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