In Gong et al. (2020), 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. (2017). 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 H(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.
G. Chen et al., "A New Global Divergence Free and Pressure-Robust Hdg Method for Tangential Boundary Control of Stokes Equations," Computer Methods in Applied Mechanics and Engineering, vol. 405, article no. 115837, Elsevier, Feb 2023.
The definitive version is available at https://doi.org/10.1016/j.cma.2022.115837
Mathematics and Statistics
Keywords and Phrases
Dirichlet Optimal Control; Hybridizable Discontinuous Galerkin Method; Pressure-Robust Method; Stokes System
International Standard Serial Number (ISSN)
Article - Journal
© 2023 Elsevier, All rights reserved.
15 Feb 2023