# What spatial discretizations work for incompressible flow with anisotropic boundary meshes?

High Reynolds number flows produce very thin boundary layers. If wall resolution is used in Large Eddy Simulation, the aspect ratio may be on the order of $10^6$. Many methods become unstable in this regime because the inf-sup constant degrades as the square root of the aspect ratio or worse. The inf-sup constant is important because it affects the condition number of the linear system and the approximation properties of the discrete solution. In particular, the following a priori bounds on the discrete error hold (Brezzi and Fortin 1991)

$$\begin{split} \mu \lVert {\mathbf u} - \mathbf u_h \rVert_{H^1} \le C \left[ \frac{\mu}{\beta} \inf_{\mathbf v \in \mathcal V} \lVert{\mathbf u - \mathbf v}\rVert_{H^1} + \inf_{q \in \mathcal Q} \lVert p-q \rVert_{L^2} \right] \\ \lVert{p - p_h}\rVert_{L^2} \le \frac{C}{\beta} \left[ \frac{\mu}{\beta} \inf_{\mathbf v \in \mathcal V} \lVert{\mathbf u - \mathbf v}\rVert_{H^1} + \inf_{q\in \mathcal Q} \lVert{p-q}\rVert_{L^2} \right] \end{split}$$

where $\mu$ is the dynamic viscosity and $\beta$ is the inf-sup constant. From this we see that as $\beta \to 0$, the velocity and (especially) pressure approximations becomes worse than the best available in the finite element space (i.e. the constant of Galerkin optimality grows as $\beta^{-1}$ and $\beta^{-2}$ respectively).

What methods have uniform inf-sup stability independent of the aspect ratio?

Which of these can be used with unstructured meshes?

How do the estimates generalize to high order approximations?

MAC finite difference schemes (Harlow and Welch 1965) are uniformly stable, but require smooth structured grids and are only second order accurate.

Finite element methods are preferred for unstructured and high order methods. For continuous Galerkin finite element methods, there are no known spaces that have optimal approximation properties and are uniformly stable.

• $Q_k-P_{k-1}^{\mathrm{disc}}$ has optimal approximation properties and is locally conservative, but the inf-sup constant degrades as the square root of the aspect ratio. See Bernardi & Maday 1999 for details.

• $Q_k-Q_{k-2}^{\mathrm{disc}}$ has inf-sup constant independent of the aspect ratio and is locally conservative, but the inf-sup constant scales as $\mathcal{O}\left(k^{\frac{1-d}{2}}\right)$ as the polynomial order is increased (Maday et al. 1992) on shape-regular meshes. On meshes with hanging nodes or reentrant corners, this bound is sharp in 2D (Schoetzau et al 1998), but further degrades to $k^{-3/2}$ in 3D (Toselli & Schwab 2003).

• The rotated $Q_1-P_0$ nonconforming element from Rannacher & Turek 1994 is uniformly stable, has optimal approximation properties, and is locally conservative, but it does not satisfy a discrete Korn's inequality, so it needs boundary corrections for some boundary conditions and cannot be used for variable viscosity flows. Subsequent work by the authors has saught to stabilize these methods using edge fluxes, but the resulting discretizations lose many of the attractive efficiency properties.

• Ainsworth and Coggins 2000 construct highly technical spaces that do somewhat better, but seem to be of limited utility.

For discontinuous Galerkin, the picture is somewhat better:

• The discontinuous space $Q_k-Q_{k-1}$ is uniformly stable and has optimal approximation properties (Schoetzau, Schwab, and Toselli 2004). This combination is not available with continuous velocity spaces. The inf-sup constant is still dependent on polynomial degree, however, scaling as $k^{-3/2}$.