Analysis of pressure-robust embedded-hybridized discontinuous Galerkin methods for the Stokes problem under minimal regularity

TL;DR

This paper analyzes pressure-robust hybridizable discontinuous Galerkin methods for the Stokes problem, deriving optimal error estimates under minimal regularity assumptions and validating them through numerical experiments.

Contribution

It provides the first a priori error estimates for these methods under minimal regularity, demonstrating pressure robustness and optimal convergence.

Findings

Error estimates for velocity are independent of pressure.

Methods achieve optimal convergence in a discrete energy norm.

Numerical results confirm theoretical predictions.

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 $H(\textrm{div})$-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.