Preconditioning a Hybridizable Discontinuous Galerkin Method for Navier–Stokes at High Reynolds Number

Abstract. We introduce a preconditioner for a hybridizable discontinuous Galerkin discretization of the linearized Navier–Stokes equations at high Reynolds number. The preconditioner is based on an augmented Lagrangian (AL) approach of the full discretization. Unlike standard grad-div type augmentation, however, we consider augmentation based on divergence-conformity. We introduce a new algebraic analysis of AL-type preconditioning in the context of singular perturbations. With this augmentation we introduce two different, well-conditioned, and easy to solve matrices to approximate the trace pressure Schur complement. Numerical examples demonstrate that these approximations to the trace pressure Schur complement are highly robust in mesh spacing and Reynolds number. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/idaholab/moose and in the supplementary materials ( moose-next.zip [242MB]). [Formula: see text]