Bilinear optimal control for the Stokes-Brinkman equations: a priori and a posteriori error analyses

TL;DR

This paper studies a bilinear optimal control problem for the Stokes-Brinkman equations, providing a comprehensive analysis including existence, optimality conditions, and error estimates for finite element discretizations.

Contribution

It introduces two finite element methods for the problem, analyzes their convergence, and develops a posteriori error estimators, advancing numerical analysis for these control problems.

Findings

Existence and optimality conditions established for the control problem.

Convergence and a priori error estimates derived for both discretization schemes.

A posteriori error estimators with reliability and efficiency bounds developed.

Abstract

We analyze a bilinear optimal control problem for the Stokes--Brinkman equations: the control variable enters the state equations as a coefficient. In two- and three-dimensional Lipschitz domains, we perform a complete continuous analysis that includes the existence of solutions and first- and second-order optimality conditions. We also develop two finite element methods that differ fundamentally in whether the admissible control set is discretized or not. For each of the proposed methods, we perform a convergence analysis and derive a priori error estimates; the latter under the assumption that the domain is convex. Finally, assuming that the domain is Lipschitz, we develop an a posteriori error estimator for each discretization scheme, obtain a global reliability bound, and investigate local efficiency estimates.