Error estimate of a decoupled numerical scheme for the Cahn–Hilliard–Stokes–Darcy system
We analyze a fully discrete finite element numerical scheme for the Cahn-Hilliard-Stokes-Darcy system that models two-phase flows in coupled free flow and porous media. To avoid a well-known difficulty associated with the coupling between the Cahn-Hilliard equation and the fluid motion, we make use of the operator-splitting in the numerical scheme, so that these two solvers are decoupled, which in turn would greatly improve the computational efficiency. The unique solvability and the energy stability have been proved in Chen et al. (2017, Uniquely solvable and energy stable decoupled numerical schemes for the Cahn-Hilliard-Stokes-Darcy system for two-phase flows in karstic geometry. Numer. Math., 137, 229-255). In this work, we carry out a detailed convergence analysis and error estimate for the fully discrete finite element scheme, so that the optimal rate convergence order is established in the energy norm, i.e., in the l(infinity) (0, T; H1)boolean AND l(2)(0, T; H-2) norm for the phase variables, as well as in the l(infinity) (0, T; H1)boolean AND l(2)(0, T; H-2) norm for the velocity variable. Such an energy norm error estimate leads to a cancelation of a nonlinear error term associated with the convection part, which turns out to be a key step to pass through the analysis. In addition, a discrete l(2)(0; T; H-3) bound of the numerical solution for the phase variables plays an important role in the error estimate, which is accomplished via a discrete version of Gagliardo-Nirenberg inequality in the finite element setting.
National Key R&D Program of China[2019YFA0709502] ; National Science Foundation of China[12071090,11871159] ; National Science Foundation[
|WOS Research Area|
|WOS Accession No|
|EI Accession Number|
Computational efficiency ; Estimation ; Finite element method ; Porous materials ; Two phase flow
|ESI Classification Code|
Fluid Flow, General:631.1 ; Mathematics:921 ; Numerical Methods:921.6 ; Materials Science:951
|ESI Research Field|
Cited Times [WOS]:8
|Document Type||Journal Article|
|Department||Department of Mathematics|
1.School of Mathematical Sciences, Fudan University, Shanghai 200433, China
2.Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USA
3.Department of Mathematics, University of Massachusetts Dartmouth, North Dartmouth, MA 02747, USA
4.Department of Mathematics, SUSTech International Center for Mathematics, National Center for Applied Mathematics Shenzhen, Guangdong Provincial Key Laboratory of Computational Sicience and Material Design, Southern University of Science and Technology, Shenzhen 518055, China
Wenbin,Chen,Shufen,Wang,Yichao,Zhang,等. Error estimate of a decoupled numerical scheme for the Cahn–Hilliard–Stokes–Darcy system[J]. IMA JOURNAL OF NUMERICAL ANALYSIS,2021,0(0):1-35.
Wenbin,Chen,Shufen,Wang,Yichao,Zhang,Daozhi,Han,Cheng,Wang,&Xiaoming,Wang.(2021).Error estimate of a decoupled numerical scheme for the Cahn–Hilliard–Stokes–Darcy system.IMA JOURNAL OF NUMERICAL ANALYSIS,0(0),1-35.
Wenbin,Chen,et al."Error estimate of a decoupled numerical scheme for the Cahn–Hilliard–Stokes–Darcy system".IMA JOURNAL OF NUMERICAL ANALYSIS 0.0(2021):1-35.
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