We propose and analyze a finite element approximation of the relaxed Cahn–Hilliard equation [Perthame and Poulain, Eur. J. Appl. Math. 32 (2021) 89–112.] 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. We propose two different time discretizations leading to a non-linear and a linear scheme. Moreover, we show the well-posedness and convergence of solutions of the non-linear numerical scheme. Finally, we validate our scheme by presenting numerical simulations in one and two dimensions.
Keywords: Degenerate Cahn–Hilliard equation, single-well potential, continuous Galerkin finite elements, upwind scheme, convergence analysis
@article{M2AN_2022__56_5_1741_0,
author = {Bubba, Federica and Poulain, Alexandre},
title = {A nonnegativity preserving scheme for the relaxed {Cahn{\textendash}Hilliard} equation with single-well potential and degenerate mobility},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1741--1772},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {5},
doi = {10.1051/m2an/2022050},
mrnumber = {4454158},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2022050/}
}
TY - JOUR AU - Bubba, Federica AU - Poulain, Alexandre TI - A nonnegativity preserving scheme for the relaxed Cahn–Hilliard equation with single-well potential and degenerate mobility JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2022 SP - 1741 EP - 1772 VL - 56 IS - 5 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2022050/ DO - 10.1051/m2an/2022050 LA - en ID - M2AN_2022__56_5_1741_0 ER -
%0 Journal Article %A Bubba, Federica %A Poulain, Alexandre %T A nonnegativity preserving scheme for the relaxed Cahn–Hilliard equation with single-well potential and degenerate mobility %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2022 %P 1741-1772 %V 56 %N 5 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2022050/ %R 10.1051/m2an/2022050 %G en %F M2AN_2022__56_5_1741_0
Bubba, Federica; Poulain, Alexandre. A nonnegativity preserving scheme for the relaxed Cahn–Hilliard equation with single-well potential and degenerate mobility. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 5, pp. 1741-1772. doi: 10.1051/m2an/2022050
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