Inf-sup condition for Stokes with outflow condition

TL;DR

This paper extends the inf-sup condition for the Stokes equations to domains with outflow boundary conditions, addressing the challenge posed by non-zero mean pressure functions.

Contribution

It derives a new form of the inf-sup condition applicable to domains with outflow boundaries, where pressure functions do not necessarily have zero mean.

Findings

Established the inf-sup condition for outflow boundary cases

Provided theoretical framework for pressure functions without zero mean

Enhanced the mathematical understanding of Stokes problems with outflow conditions

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 $p\in L^2(\Omega)\setminus \mathbb{R}$, (i.e. with mean value zero) a velocity $u\in H^1_0(\Omega)^d$ 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 $\Gamma_N\subset\partial\Omega$, the inf-sup condition cannot be used in this form, since the pressure here comes from $L^2(\Omega)$ and does not necessarily have zero mean value. In this note, we derive the inf-sup condition for the case of outflow boundaries.