Robust preconditioning for an HDG discretization of the time-dependent Stokes equations
This work addresses the computational efficiency of solving time-dependent Stokes problems for researchers in numerical analysis and computational fluid dynamics, but it is incremental as it builds upon previous theoretical frameworks.
The authors tackled the problem of developing parameter-robust preconditioners for linear systems from a hybridizable discontinuous Galerkin discretization of time-dependent Stokes equations, resulting in new preconditioners that remain robust with respect to all physical and discretization parameters, as verified by numerical experiments in 2D and 3D.
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.