The paper analyzes the Morley element method for the Cahn–Hilliard equation. The objective is to prove the numerical interfaces of the Morley element method approximate the Hele-Shaw flow. It is achieved by establishing the optimal error estimates which depend on polynomially, and the error estimates should be established from lower norms to higher norms progressively. If the higher norm error bound is derived by choosing test function directly, we cannot obtain the optimal error order, and we cannot establish the error bound which depends on polynomially either. Different from the discontinuous Galerkin (DG) space [Feng et al. SIAM J. Numer. Anal. 54 (2016) 825–847], the Morley element space does not contain the finite element space as a subspace such that the projection theory does not work. The enriching theory is used in this paper to overcome this difficulty, and some nonstandard techniques are combined in the process such as the a priori estimates of the exact solution u, integration by parts in space, summation by parts in time, and special properties of the Morley elements. If one of these techniques is lacked, either we can only obtain the sub-optimal piecewise L∞(H2) error order, or we can merely obtain the error bounds which are exponentially dependent on . Numerical results are presented to validate the optimal L∞(H2) error order and the asymptotic behavior of the solutions of the Cahn–Hilliard equation.
Mots-clés : Morley element, Cahn–Hilliard equation, generalized coercivity result, 1/ε-polynomial dependence, Hele-Shaw flow
@article{M2AN_2020__54_3_1025_0,
author = {Wu, Shuonan and Li, Yukun},
title = {Analysis of the {Morley} element for the {Cahn{\textendash}Hilliard} equation and the {Hele-Shaw} flow},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1025--1052},
publisher = {EDP-Sciences},
volume = {54},
number = {3},
year = {2020},
doi = {10.1051/m2an/2019085},
mrnumber = {4091855},
zbl = {1437.65204},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2019085/}
}
TY - JOUR AU - Wu, Shuonan AU - Li, Yukun TI - Analysis of the Morley element for the Cahn–Hilliard equation and the Hele-Shaw flow JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2020 SP - 1025 EP - 1052 VL - 54 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2019085/ DO - 10.1051/m2an/2019085 LA - en ID - M2AN_2020__54_3_1025_0 ER -
%0 Journal Article %A Wu, Shuonan %A Li, Yukun %T Analysis of the Morley element for the Cahn–Hilliard equation and the Hele-Shaw flow %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2020 %P 1025-1052 %V 54 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2019085/ %R 10.1051/m2an/2019085 %G en %F M2AN_2020__54_3_1025_0
Wu, Shuonan; Li, Yukun. Analysis of the Morley element for the Cahn–Hilliard equation and the Hele-Shaw flow. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 3, pp. 1025-1052. doi : 10.1051/m2an/2019085. http://www.numdam.org/articles/10.1051/m2an/2019085/
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