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

TL;DR

This paper develops finite element methods with error estimators for an optimal control problem governed by steady Navier-Stokes-Brinkman equations, enabling efficient and accurate flow control analysis.

Contribution

It introduces three finite element discretization schemes with a posteriori error estimators for the Navier-Stokes-Brinkman control problem, including convergence analysis and adaptive refinement strategies.

Findings

Optimal convergence rates demonstrated through numerical experiments.

Residual-based a posteriori error estimators effectively guide adaptive mesh refinement.

Framework provides reliable insights into flow control mechanisms.

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 $\gamma$ 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.