Pressure-Robust $H(\mathrm{div})$-Conforming HDG Methods for the Steady Stokes Equations with an Application to Tangential Boundary Control

TL;DR

This paper introduces pressure-robust $H( ext{div})$-conforming HDG methods for steady Stokes equations, achieving optimal convergence and divergence-free velocities, with applications to tangential boundary control.

Contribution

It develops a pressure-robust HDG framework that does not require high pressure regularity and provides rigorous error analysis and solver insights.

Findings

Optimal energy-norm and $L^2$-velocity convergence for BDM variants.

Optimal velocity convergence and weaker pressure estimates for RT variants.

Numerical experiments confirm convergence rates, divergence-free velocities, and robustness.

Abstract

We develop a family of $H(\mathrm{div})$-conforming hybridizable discontinuous Galerkin methods for the steady Stokes equations based on BDM and RT velocity spaces with either discontinuous or continuous hybrid traces. In contrast to our earlier pressure-robust HDG method for tangential boundary control, the present analysis does not require the pressure to belong to $H^1$; instead, the consistency argument only assumes low pressure regularity. The discrete velocities are exactly divergence-free, which yields pressure robustness. For the BDM variants we derive optimal energy-norm estimates and optimal $L^2$-velocity convergence, while for the RT variants we obtain optimal velocity convergence and weaker pressure estimates. We also analyze the hybridized linear system and prove a uniform spectral equivalence for the pressure Schur complement relevant to iterative solvers. As an application, we revisit the Stokes tangential boundary control problem and derive error estimates for the control, state, and adjoint variables using the BDM discontinuous-trace scheme. Two- and three-dimensional numerical experiments confirm the predicted convergence rates, the exact divergence-free property, and the robustness of the method with respect to the viscosity parameter.