An advection-robust Hybrid High-Order method for the Oseen problem
Provides a theoretical framework for robust numerical solutions of the Oseen problem, which is relevant for computational fluid dynamics, though the contribution is incremental as it extends existing HHO methods.
This work develops advection-robust Hybrid High-Order discretizations for the Oseen equations, proving energy error estimates that scale as O(h^{k+1}) in diffusion-dominated regimes and O(h^{k+1/2}) in advection-dominated regimes, with intermediate powers otherwise.
In this work, we study advection-robust Hybrid High-Order discretizations of the Oseen equations. For a given integer $k\ge 0$, the discrete velocity unknowns are vector-valued polynomials of total degree $\le k$ on mesh elements and faces, while the pressure unknowns are discontinuous polynomials of total degree $\le k$ on the mesh. From the discrete unknowns, three relevant quantities are reconstructed inside each element: a velocity of total degree $\le(k+1)$, 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 an $\mathcal{O}(h_T^{k+1})$-term in the diffusion-dominated regime, an $\mathcal{O}(h_T^{k+\frac12})$-term in the advection-dominated regime, and scales with intermediate powers of $h_T$ in between. Numerical results complete the exposition.