A Reynolds-semi-robust and pressure-robust Hybrid High-Order method for the time dependent incompressible Navier-Stokes equations on general meshes

In this work we develop and analyze a Reynolds-semi-robust and pressure-robust Hybrid High-Order (HHO) discretization of the incompressible Navier-Stokes equations. Reynolds-semi- robustness refers to the fact that, under suitable regularity assumptions, the right-hand side of the velocity error estimate does not depend on the inverse of the viscosity. This property is obtained here through a penalty term which involves a subtle projection of the convective term on a subgrid space constructed element by element. Moreover, a method has the pressure- robustness property when it guarantees velocity error estimates that are independent of the pressure. The estimated convergence order for the L infinity ( L 2 )- and L 2 ( energy )- norm of the velocity is h k + 2 1 , which matches the best results for continuous and discontinuous Galerkin methods and corresponds to the one expected for HHO methods in convection-dominated regimes. Two-dimensional numerical results on a variety of polygonal meshes complete the exposition.