A posteriori error estimators for stabilized finite element approximations of an optimal control problem
This work provides practical error control for adaptive algorithms in optimal control problems, which is valuable for computational scientists solving PDE-constrained optimization.
The paper derives fully computable a posteriori error estimators for optimal control problems governed by convection-reaction-diffusion equations with control constraints, using stabilized finite element methods. The estimators are robust and include all constants, enabling their use as stopping criteria in adaptive algorithms, with numerical examples in 2D and 3D.
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.