An Advection-Robust Hybrid High-Order Method for the Oseen Problem

In this work, we study advection-robust Hybrid High-Order discretizations of the Oseen equations. For a given integer k >= 0, the discrete velocity unknowns are vector-valued polynomials of total degree <= k on mesh elements and faces, while the pressure unknowns are discontinuous polynomials of total degree <= k on the mesh. From the discrete unknowns, three relevant quantities are reconstructed inside each element: a velocity of total degree <= (k + 1), a discrete advective derivative, and a discrete divergence. These reconstructions are used to formulate the discretizations of the viscous, advective, and velocity-pressure coupling terms, respectively. Well-posedness is ensured through appropriate high-order stabilization terms. We prove energy error estimates that are advection-robust for the velocity, and show that each mesh element T of diameter h(T) contributes to the discretization error with anO(h(T)(k+1))-term in the diffusion-dominated regime, anO(h(T)(k+1/2))-term in the advection-dominated regime, and scales with intermediate powers of hT in between. Numerical results complete the exposition.