TY - JOUR
T1 - Phase-field dynamics with transfer of materials
T2 - The Cahn-Hillard equation with reaction rate dependent dynamic boundary conditions
AU - Knopf, Patrik
AU - Lam, Kei Fong
AU - Liu, Chun
AU - Metzger, Stefan
N1 - Publisher Copyright:
© EDP Sciences, SMAI 2021.
PY - 2021/2/18
Y1 - 2021/2/18
N2 - The Cahn-Hilliard equation is one of the most common models to describe phase separation processes of a mixture of two materials. For a better description of short-range interactions between the material and the boundary, various dynamic boundary conditions for the Cahn-Hilliard equation have been proposed and investigated in recent times. Of particular interests are the model by Goldstein et al. [Phys. D 240 (2011) 754-766] and the model by Liu and Wu [Arch. Ration. Mech. Anal. 233 (2019) 167-247]. Both of these models satisfy similar physical properties but differ greatly in their mass conservation behaviour. In this paper we introduce a new model which interpolates between these previous models, and investigate analytical properties such as the existence of unique solutions and convergence to the previous models mentioned above in both the weak and the strong sense. For the strong convergences we also establish rates in terms of the interpolation parameter, which are supported by numerical simulations obtained from a fully discrete, unconditionally stable and convergent finite element scheme for the new interpolation model.
AB - The Cahn-Hilliard equation is one of the most common models to describe phase separation processes of a mixture of two materials. For a better description of short-range interactions between the material and the boundary, various dynamic boundary conditions for the Cahn-Hilliard equation have been proposed and investigated in recent times. Of particular interests are the model by Goldstein et al. [Phys. D 240 (2011) 754-766] and the model by Liu and Wu [Arch. Ration. Mech. Anal. 233 (2019) 167-247]. Both of these models satisfy similar physical properties but differ greatly in their mass conservation behaviour. In this paper we introduce a new model which interpolates between these previous models, and investigate analytical properties such as the existence of unique solutions and convergence to the previous models mentioned above in both the weak and the strong sense. For the strong convergences we also establish rates in terms of the interpolation parameter, which are supported by numerical simulations obtained from a fully discrete, unconditionally stable and convergent finite element scheme for the new interpolation model.
KW - Cahn-Hilliard equation
KW - Dynamic boundary conditions
KW - Finite element analysis
KW - Gradient flow
KW - Relaxation by Robin boundary conditions
UR - http://www.scopus.com/inward/record.url?scp=85101242335&partnerID=8YFLogxK
U2 - 10.1051/m2an/2020090
DO - 10.1051/m2an/2020090
M3 - Journal article
AN - SCOPUS:85101242335
SN - 2822-7840
VL - 55
SP - 229
EP - 282
JO - ESAIM: Mathematical Modelling and Numerical Analysis
JF - ESAIM: Mathematical Modelling and Numerical Analysis
IS - 1
ER -