Error estimate of a decoupled numerical scheme for the Cahn-Hilliard-Stokes-Darcy system

TL;DR

This paper provides a detailed convergence analysis and error estimate for a decoupled finite element scheme solving the coupled Cahn-Hilliard-Stokes-Darcy system, demonstrating optimal convergence rates and stability in energy norms.

Contribution

It introduces a decoupled numerical scheme with proven convergence and optimal error estimates for the complex coupled system, improving computational efficiency.

Findings

Optimal convergence order in energy norm for phase variables.

Energy stability and unique solvability of the scheme.

Key nonlinear error cancellation achieved in analysis.

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~\cite{CHW2017}. 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 cancellation 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.