An advection-robust Hybrid High-Order method for the Oseen problem

TL;DR

This paper introduces an advection-robust Hybrid High-Order discretization method for the Oseen equations, providing high-order accuracy and stability across different flow regimes, validated by theoretical error estimates and numerical tests.

Contribution

It develops a novel Hybrid High-Order method with robust advection handling and proven error estimates for the Oseen problem, advancing numerical fluid dynamics techniques.

Findings

Energy error estimates are advection-robust.

Discretization error scales with mesh size as h^{k+1} in diffusion regime.

Error scales as h^{k+1/2} in advection-dominated regime.

Abstract

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.