Inf-sup condition for Stokes with outflow condition
Malte Braack, Thomas Richter
TL;DR
This work extends the classical inf-sup (Ladyzhenskaya–Babuška–Brezzi) stability result for the Stokes system to domains with outflow boundaries where the pressure may have nonzero mean. It provides a constructive proof by decomposing the pressure as $p=\bar p+p_0$, applying the standard inf-sup on $H^1_0(\Omega)^d \times L^2_0(\Omega)$ to obtain a velocity $\boldsymbol u_0$, and building a localized velocity $\bar{\boldsymbol u}$ near a boundary point on $\Gamma_N$ to control the mean pressure component; combining these yields an explicit lower bound for the inf-sup constant $\gamma$. The main result shows $\inf_{p\in Q\setminus\{0\}} \sup_{\boldsymbol u\in \boldsymbol V\setminus\{0\}} \frac{(\operatorname{div}\boldsymbol u,p)}{\|\boldsymbol u\|_{\boldsymbol V}\|p\|} \ge \gamma > 0$ with $\gamma = \frac{\gamma_0}{2}\min\left\{1,\frac{c_1}{d c_2 |\Omega|^{1/2}}\right\}\left(1+\frac{\gamma_0}{2d}\right)^{-1}$. This yields quantitative scaling insights for common geometries (e.g., channels and balls) and informs stable numerical discretizations for Stokes flow with do-nothing outflow boundaries. The work also highlights future directions, such as incorporating pressure fixation on $\Gamma_N$ to tighten the bound.
Abstract
The inf-sup condition is one of the essential tools in the analysis of the Stokes equations and especially in numerical analysis. In its usual form, the condition states that for every pressure $p\in L^2(Ω)\setminus \mathbb{R}$, (i.e. with mean value zero) a velocity $u\in H^1_0(Ω)^d$ can be found, so that $(div\,u,p)=\|p\|^2$ and $\|\nabla u\|\le c \|p\|$ applies, where $c>0$ does not depend on $u$ and $p$. However, if we consider domains that have a Neumann-type outflow condition on part of the boundary $Γ_N\subset\partialΩ$, the inf-sup condition cannot be used in this form, since the pressure here comes from $L^2(Ω)$ and does not necessarily have zero mean value. In this note, we derive the inf-sup condition for the case of outflow boundaries.