A posteriori error estimation for finite element approximations of a PDE-constrained optimization problem in fluid dynamics

TL;DR

This paper develops reliable a posteriori error estimators for finite element solutions of PDE-constrained optimization problems in fluid dynamics, applicable to various stabilized and standard finite element methods, with guaranteed error bounds.

Contribution

It introduces a general framework for a posteriori error estimation in PDE-constrained optimization, accommodating stabilized methods without requiring specific stabilization relations.

Findings

Error estimators are globally reliable and locally efficient.

The estimators provide guaranteed upper bounds on the error.

Numerical examples validate the theoretical results.

Abstract

We derive globally reliable a posteriori error estimators for a PDE-constrained optimization problem involving linear models in fluid dynamics as state equation; control constraints are also considered. The corresponding local error indicators are locally efficient. The assumptions under which we perform the analysis are such that they can be satisfied for a wide variety of stabilized finite element methods as well as for standard finite element methods. When stabilized methods are considered, no a priori relation between the stabilization terms for the state and adjoint equations is required. If a lower bound for the inf-sup constant is available, a posteriori error estimators that are fully computable and provide guaranteed upper bounds on the norm of the error can be obtained. We illustrate the theory with numerical examples.