Pressure-Robust Staggered DG Methods for the Navier–Stokes Equations on General Meshes

Abstract In this paper, we design and analyze staggered discontinuous Galerkin methods of arbitrary polynomial orders for the stationary Navier–Stokes equations on polygonal meshes. The exact divergence-free condition for the velocity is satisfied thanks to the carefully designed finite element spaces. The resulting method is pressure-robust so that the pressure approximation does not influence the velocity approximation, which is highly appreciated from a practical point of view. An edge-wise stabilization is proposed for the nonlinear convective term, which preserves the non-negativity. The optimal convergence estimates for all the variables in the $$L^2$$ L 2 norm are proved. For a small enough rotational force, the velocity error is independent of the Reynolds number and of the pressure. Superconvergence can be achieved for the velocity error under a suitable projection. Numerical experiments are provided to validate the theoretical findings and demonstrate the performance of the proposed method.