A Posteriori Error Estimation Improved by a Reconstruction Operator for the Stokes Optimal Control Problem

TL;DR

This paper introduces an improved a posteriori error estimator for the Stokes optimal control problem that uses a divergence-free reconstruction operator, enabling pressure-independent error estimation and better separation of velocity and pressure errors.

Contribution

The paper develops a new residual-based a posteriori error estimator that is pressure-robust and separates velocity and pressure errors in the context of Stokes optimal control.

Findings

The estimator is globally reliable and efficient.

Numerical experiments validate the theoretical results.

Velocity error estimates are independent of pressure.

Abstract

This paper focuses on a posteriori error estimates for a pressure-robust finite element method, which incorporates a divergence-free reconstruction operator, within the context of the distributed optimal control problem constrained by the Stokes equations. We develop an enhanced residual-based a posteriori error estimator that is independent of pressure and establish its global reliability and efficiency. The proposed a posteriori error estimator enables the separation of velocity and pressure errors in a posteriori error estimation, ensuring velocity-related estimates are free of pressure influence. Numerical experiments confirm our conclusions.