A posteriori error estimators for stabilized finite element approximations of an optimal control problem

TL;DR

This paper develops fully computable a posteriori error estimators for stabilized finite element methods applied to an optimal control problem governed by convection-reaction-diffusion equations, enabling adaptive solution refinement.

Contribution

It introduces a fully specified, robust a posteriori error estimator for stabilized finite element approximations of an optimal control problem with control constraints.

Findings

Estimator is fully computable with explicit constants.

Numerical examples demonstrate estimator effectiveness in 2D and 3D.

Suitable for adaptive algorithms in optimal control simulations.

Abstract

We derive a posteriori error estimators for an optimal control problem governed by a convection-reaction-diffusion equation; control constraints are also considered. We consider a family of low-order stabilized finite element methods to approximate the solutions to the state and adjoint equations. We obtain a fully computable a posteriori error estimator for the optimal control problem. All the constants that appear in the upper bound for the error are fully specified. Therefore, the proposed estimator can be used as a stopping criterion in adaptive algorithms. We also obtain a robust a posteriori error estimator for when the error is measured in a norm that involves the dual norm of the convective derivative. Numerical examples, in two and three dimensions, are presented to illustrate the theory.