Robust preconditioning for an HDG discretization of the time-dependent Stokes equations

TL;DR

This paper develops parameter-robust preconditioners for linear systems from HDG discretizations of the time-dependent Stokes equations, ensuring stability across all physical and discretization parameters.

Contribution

It extends previous theoretical frameworks to derive new preconditioners that are robust with respect to all parameters in the HDG discretization of the Stokes problem.

Findings

Preconditioners are robust across all physical and discretization parameters.

Numerical experiments confirm theoretical robustness in 2D and 3D.

Uniform well-posedness of the HDG formulation is established.

Abstract

We present parameter-robust preconditioners for linear systems that arise after applying static condensation to a hybridizable discontinuous Galerkin (HDG) discretization of the time-dependent Stokes problem. Building upon the theoretical framework introduced in our previous work [SIAM Journal on Scientific Computing, 47(6):A3212-A3238, 2025], we extend the analysis to derive new preconditioners that remain robust with respect to all physical and discretization parameters. The construction relies on first establishing uniform well-posedness of the HDG formulation (before static condensation) through appropriately defined norms. Based on this result, we identify sufficient conditions that a norm on the face space must satisfy to guarantee parameter-robustness of the resulting preconditioner for the statically condensed HDG system. Numerical experiments in two and three dimensions verify our theoretical results.