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In this work, we study advection-robust Hybrid High-Order discretizations of the Oseen equations. For a given integer $$k\geqslant 0$$k⩾0, the discrete velocity unknowns are vector-valued polynomials of total degree $$\leqslant \, k$$⩽k on mesh elements and faces, while the pressure unknowns are discontinuous polynomials of total degree $$\leqslant \,k$$⩽k on the mesh. From the discrete unknowns, three relevant quantities are reconstructed inside each element: a velocity of total degree $$\leqslant \,(k+1)$$⩽(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 $$h_T$$hT contributes to the discretization error with an $$\mathcal {O}(h_{T}^{k+1})$$O(hTk+1)-term in the diffusion-dominated regime, an $$\mathcal {O}(h_{T}^{k+\frac{1}{2}})$$O(hTk+12)-term in the advection-dominated regime, and scales with intermediate powers of $$h_T$$hT in between. Numerical results complete the exposition.

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