A finite element scheme for an optimal control problem on steady Navier-Stokes-Brinkman equations

TL;DR

This work tackles identifying the permeability parameter $\gamma$ in a steady Navier–Stokes–Brinkman model from velocity observations by formulating a PDE-constrained optimal control problem. It develops a rigorous finite element framework employing three discretization schemes (fully discrete with discontinuous or continuous controls and a semi-discrete variational scheme) and derives both a priori error estimates and residual-based a posteriori estimators to drive adaptive mesh refinement. Reliability and efficiency of the estimators are established, first- and second-order optimality conditions are analyzed, and numerical experiments with manufactured solutions and obstacle-recovery scenarios validate optimal convergence rates and estimator effectiveness. The approach reduces computational costs while maintaining accuracy, offering a robust tool for porous media flow control and paving the way for extensions to time-dependent, stochastic, or multiphysics settings, with a practical FEniCS implementation demonstrated.

Abstract

This paper presents a rigorous finite element framework for solving an optimal control problem governed by the steady Navier-Stokes-Brinkman equations, focusing on identifying a scalar permeability parameter $γ$ from local velocity observations. Three different finite element discretization schemes are proposed, and a priori error estimates are proven under appropriate regularity assumptions for each one. A key contribution of this paper is the development of residual-based a posteriori error estimators for both fully discrete and semi-discrete schemes, guiding adaptive mesh refinement to achieve comparable accuracy with fewer degrees of freedom. The method of manufactured solutions is used for numerical experiments to validate the theoretical findings, to demonstrate optimal convergence rates and the effectivity index is evaluated to measure their reliability. The framework offers insights into flow control mechanisms and paving the way for extensions to time-dependent, stochastic, or multiphysics problems.

Figures (10)

• Figure 1: History of convergence for $\mathcal{A}^h=\mathcal{A}\cap G_0 ^{h}$ (left), $\mathcal{A}^h=\mathcal{A}\cap G_1 ^{h}$ (center) and semi-discrete schemes (right).
• Figure 2: History of convergence for $\mathbb{P}_0$ (left), $\mathbb{P}_1$ (center) and semi-discrete schemes (right).
• Figure 3: Initial mesh for adaptive refinement
• Figure 4: Adapted meshes for $\mathbb{P}_0$ (left, stage 24, 497218 dof, 49634 elements), $\mathbb{P}_1$ (center, stage 27, 440609 dof, 46119 elements) and semi-discrete schemes (right, stage 31, 451780 dof, 49898 elements).
• Figure 5: Isovalues of $\gamma^h$ on the adapted meshes for $\mathbb{P}_0$, $\mathbb{P}_1$ and semi-discrete schemes (from left to right).

Theorems & Definitions (91)

• Definition 1
• Definition 2
• Lemma 3
• Theorem 4
• proof
• Definition 5
• Definition 6
• Lemma 7
• proof
• Definition 8