Pressure-robustness in the context of optimal control

TL;DR

This paper investigates how pressure-robust discretizations improve the accuracy of optimal control in incompressible flows by ensuring orthogonality of gradient forces, with theoretical analysis and numerical validation.

Contribution

It demonstrates that pressure-robust discretizations enhance optimal control accuracy by restoring orthogonality of gradient forces, supported by theoretical estimates and numerical experiments.

Findings

Pressure-robust discretizations improve control accuracy.

Orthogonality of gradient forces is crucial for precise control.

Numerical examples validate theoretical improvements.

Abstract

This paper studies the benefits of pressure-robust discretizations in the scope of optimal control of incompressible flows. Gradient forces that may appear in the data can have a negative impact on the accuracy of state and control and can only be correctly balanced if their $L^2$-orthogonality onto discretely divergence-free test functions is restored. Perfectly orthogonal divergence-free discretizations or divergence-free reconstructions of these test functions do the trick and lead to much better analytic a priori estimates that are also validated in numerical examples.