Inf-sup estimates for the Stokes problem in a periodic channel

TL;DR

This paper derives explicit estimates for the inf-sup constant in the Stokes problem within a periodic channel, revealing how geometric features influence stability constants and providing optimal bounds.

Contribution

It introduces a method to explicitly estimate the inf-sup constant based on geometric parameters, avoiding non-constructive theorems and extending results to Lipschitz domains.

Findings

Explicit dependence of the inf-sup constant on aspect ratio and geometry.

Connection between inf-sup constant and Fourier coefficient transformations.

Optimality of estimates in the Lipschitz and $C^{1,1}$ cases.

Abstract

We derive estimates of the Babu\u{s}ka-Brezzi inf-sup constant $\beta$ for two-dimensional incompressible flow in a periodic channel with one flat boundary and the other given by a periodic, Lipschitz continuous function $h$. If $h$ is a constant function (so the domain is rectangular), we show that periodicity in one direction but not the other leads to an interesting connection between $\beta$ and the unitary operator mapping the Fourier sine coefficients of a function to its Fourier cosine coefficients. We exploit this connection to determine the dependence of $\beta$ on the aspect ratio of the rectangle. We then show how to transfer this result to the case that $h$ is $C^{1,1}$ or even $C^{0,1}$ by a change of variables. We avoid non-constructive theorems of functional analysis in order to explicitly exhibit the dependence of $\beta$ on features of the geometry such as the aspect ratio, the maximum slope, and the minimum gap thickness (if $h$ passes near the substrate). We give an example to show that our estimates are optimal in their dependence on the minimum gap thickness in the $C^{1,1}$ case, and nearly optimal in the Lipschitz case.