A posteriori error estimates for an optimal control problem with a bilinear state equation

TL;DR

This paper develops and analyzes reliable a posteriori error estimators for an optimal control problem involving elliptic PDEs with bilinear state equations, considering both fully discrete and semi-discrete schemes, and demonstrates their effectiveness through adaptive methods.

Contribution

It introduces new a posteriori error estimators for bilinear elliptic control problems and proves their reliability and efficiency for adaptive finite element methods.

Findings

Error estimators are reliable and efficient in 2D and 3D domains.

Adaptive strategies achieve optimal convergence rates.

Estimators work for both fully discrete and variational discretization schemes.

Abstract

We propose and analyze a posteriori error estimators for an optimal control problem that involves an elliptic partial differential equation as state equation and a control variable that enters the state equation as a coefficient; pointwise constraints on the control variable are considered as well. We consider two different strategies to approximate optimal variables: a fully discrete scheme in which the admissible control set is discretized with piecewise constant functions and a semi-discrete scheme where the admissible control set is not discretized; the latter scheme being based on the so-called variational discretization approach. We design, for each solution technique, an a posteriori error estimator and show, in two and three dimensional Lipschitz polygonal/polyhedral domains (not necessarily convex), that the proposed error estimator is reliable and efficient. We design, based on the devised estimators, adaptive strategies that deliver optimal experimental rates of convergence for the performed numerical examples.