Error estimate of a decoupled numerical scheme for the Cahn–Hilliard–Stokes–Darcy system
Error estimate of a decoupled numerical scheme for the Cahn–Hilliard–Stokes–Darcy system
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Cahn-Hilliard-Stokes-Darcy 系统解耦数值格式的误差估计
DOI：
10.1093/imanum/drab046
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发表时间：
2021
影响因子：
2.1
通讯作者：
Yichao Zhang
中科院分区：
文献类型：
--
作者：
Wenbin Chen;Daozhi Han;Xiaoming Wang;Shufen Wang;Yichao Zhang
摘要本文分析了Cahn-Hilliard-Stokes-Darcy系统的完全离散有限元数值格式，该格式模拟了自由流动和多孔介质中的两相流动。为了避免Cahn-Hilliard方程与流体运动之间的耦合所带来的众所周知的困难，我们在数值格式中使用了算子分裂，使这两个求解器解耦，从而大大提高了计算效率。Chen et al.（2017，岩溶几何中两相流Cahn-Hilliard-Stokes-Darcy系统的唯一可解和能量稳定解耦数值格式）证明了其唯一可解性和能量稳定性。号码。数学。， 137, 229-255)。在这项工作中，我们对完全离散有限元格式进行了详细的收敛分析和误差估计，从而在能量范数中，即在相位变量的$ell ^{infty} (0， T; H^1) cap ell ^2 (0， T; H^2)$范数中，以及在速度变量的$ell ^{infty} (0， T; H^1) cap ell ^2 (0， T; H^2)$范数中建立了最优速率收敛顺序。这样的能量范数误差估计导致与对流部分相关的非线性误差项的消除，这是通过分析的关键步骤。此外，相位变量数值解的离散$ell ^2 (0；T; H^3)$界在误差估计中起着重要作用，这是通过有限元环境下离散版的Gagliardo-Nirenberg不等式来实现的。
Abstract 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 $ell ^{infty } (0, T; H^1) cap ell ^2 (0, T; H^2)$ norm for the phase variables, as well as in the $ell ^{infty } (0, T; H^1) cap ell ^2 (0, T; H^2)$ 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 $ell ^2 (0;T; H^3)$ 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.