Analysis of Pressure-Robust Embedded-Hybridized Discontinuous Galerkin Methods for the Stokes Problem Under Minimal Regularity

Abstract

We present analysis of two lowest-order hybridizable discontinuous Galerkin methods for the Stokes problem, while making only minimal regularity assumptions on the exact solution. The methods under consideration have previously been shown to produce \(H(\text {div})\)-conforming and divergence-free approximate velocities. Using these properties, we derive a priori error estimates for the velocity that are independent of the pressure. These error estimates, which assume only \(H^{1+s}\)-regularity of the exact velocity fields for any \(s \in [0, 1]\), are optimal in a discrete energy norm. Error estimates for the velocity and pressure in the \(L^2\)-norm are also derived in this minimal regularity setting. Our theoretical findings are supported by numerical computations.