A New Global Divergence Free and Pressure-Robust HDG Method for Tangential Boundary Control of Stokes Equations

TL;DR

This paper introduces a novel divergence-free, pressure-robust HDG method for tangential boundary control of Stokes equations, improving accuracy and stability over previous schemes by ensuring velocity divergence-free conditions.

Contribution

The paper develops the first HDG scheme that is divergence-free and pressure-robust for Stokes boundary control problems, enhancing solution accuracy and theoretical properties.

Findings

The new HDG method achieves optimal convergence rates.

Numerical experiments demonstrate improved performance over previous schemes.

The scheme ensures globally divergence-free velocity solutions.

Abstract

In [ESAIM: M2AN, 54(2020), 2229-2264], we proposed an HDG method to approximate the solution of a tangential boundary control problem for the Stokes equations and obtained an optimal convergence rate for the optimal control {that reflects its global regularity}. However, the error estimates depend on the pressure, and the velocity is not divergence free. The importance of pressure-robust numerical methods for fluids was addressed by John et al. [SIAM Review, 59(2017), 492-544]. In this work, we devise a new HDG method to approximate the solution of the Stokes tangential boundary control problem; the HDG method is also of independent interest for solving the Stokes equations. This scheme yields a $\mathbf{H}(\mathrm{div})$ conforming, globally divergence free, and pressure-robust solution. To the best of our knowledge, this is the first time such a numerical scheme has been obtained for an optimal boundary control problem for the Stokes equations. We also provide numerical experiments to show the performance of the new HDG method and the advantage over the non pressure-robust scheme.