Pressure-robustness in Stokes-Darcy Optimal Control Problem with reconstruction operator
Jingshi Li, Jiachuan Zhang, Ran Zhang
TL;DR
This work addresses pressure robustness for optimal control problems constrained by the Stokes-Darcy system. It introduces a reconstruction operator to enforce exact divergence-free and interface-continuity properties, reformulating both the constraint and the objective to remove pressure-induced effects on velocity errors. The authors analyze the classical discretization's pressure dependence and then develop a reconstruction-based discretization using Raviart-Thomas spaces, establishing error estimates that prove pressure-robustness with optimal convergence under regularity assumptions. The results enhance reliability of simulations in coupled free-flow and porous-media applications, highlighting a practical pathway to robust numerical control of multiphysics flows.
Abstract
This paper presents a pressure-robust discretizations, specifically within the context of optimal control problems for the Stokes-Darcy system. The study meticulously revisits the formulation of the divergence constraint and the enforcement of normal continuity at interfaces, within the framework of the mixed finite element method (FEM). The methodology involves the strategic deployment of a reconstruction operator, which is adeptly applied to both the constraint equations and the cost functional. This is complemented by a judicious selection of finite element spaces that are tailored for approximation and reconstruction purposes. The synergy of these methodological choices leads to the realization of a discretization scheme that is pressure-robust, thereby enhancing the robustness and reliability of numerical simulations in computational mathematics.