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