HDG Navier-Stokes
Interior Penalty
Interior penalty HDG Navier-Stokes is favored for advection-dominated flows due to its pressure robustness. The finite element fields include interior and facet velocities and interior and facet pressures. We have built a preconditioning strategy for IP-HDG Navier-Stokes that is extremely robust in approximating the Schur complement with respect to mesh size and Reynolds number. Its detailed description can be found in the arXiv paper. The preconditioning strategy leverages an augmented Lagrange like addition to the weak form, which allows the accurate approximation of the Schur complement. However, as also discussed in NavierStokesProblem, the addition of an augmented Lagrange term introduces a large symmetric singular perturbation in the momentum block, transferring solver difficulty from the Schur complement to the momentum block. As discussed in the arXiv paper, our current strategy is to use an inexact LU decomposition with butterfly compression for the (trace) momentum block. This is effective until reaching sufficiently large problem size and Reynolds number at which point the inexactness of the decomposition due to compression loss is unable to resolve the poor conditioning resulting from the combination of the large singular perturbation and advection dominance. We hypothesize that introduction of turbulence models, whether via RANS or subgrid scale models in LES, would add sufficient viscosity to restore the effectiveness of the inexact LU. However, this is an open question that would have to be addressed by further work.
Hybridizable Local Discontinuous Galerkin
The local discontinuous Galerkin method introduces additional finite element fields corresponding to the interior and facet velocity gradients. This method is a strong choice for diffusion-dominated flows because the velocity gradient converges with optimal order whereas for other methods, the velocity gradient convergence is suboptimal. Additionally, with postprocessing the velocity converges with an additional order, e.g. it is superconvergent. Additional information may be found at the core HDG kernel page.