Preconditioning a hybridizable discontinuous Galerkin method for Navier-Stokes at high Reynolds number

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 approach of the full discretization. Unlike standard grad-div type augmentation, however, we consider augmentation based on divergence-conformity. With this augmentation we introduce two different, well-conditioned, and easy to solve matrices to approximate the trace pressure Schur complement. To introduce a completely algebraic solver, we propose to use multifrontal sparse LU solvers using butterfly compression to solve the trace velocity block. Numerical examples demonstrate that the trace pressure Schur complement is highly robust in mesh spacing and Reynolds number and that the multifrontal inexact LU performs well for a wide range of Reynolds numbers.