Adaptive finite element method for an unregularized semilinear optimal control problem

TL;DR

This paper develops an adaptive finite element method with an a posteriori error estimator for a semilinear optimal control problem constrained by PDEs, improving solution accuracy through adaptive refinement.

Contribution

It introduces a novel a posteriori error estimator for an affine semilinear PDE-constrained optimal control problem and demonstrates its effectiveness in adaptive algorithms.

Findings

The error estimator is reliable and efficient under certain conditions.

Adaptive refinement improves solution accuracy in numerical examples.

The method effectively handles control constraints and nonlinear PDEs.

Abstract

We devise an a posteriori error estimator for an affine optimal control problem subject to a semilinear elliptic PDE and control constraints. To approximate the problem, we consider a semidiscrete scheme based on the variational discretization approach. For this solution technique, we design an a posteriori error estimator that accounts for the discretization of the state and adjoint equations, and prove, under suitable local growth conditions of optimal controls, reliability and efficiency properties of such error estimator. A simple adaptive strategy based on the devised estimator is designed and its performance is illustrated with numerical examples.