Inf-sup estimates for the Stokes problem in a periodic channel
| dc.creator | Wilkening, Jon | |
| dc.date | 2007-06-27 | |
| dc.date.accessioned | 2026-07-07T08:12:44Z | |
| dc.date.available | 2026-07-07T08:12:44Z | |
| dc.description | We derive estimates of the Babuuska-Brezzi inf-sup constant $β$ 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 $β$ 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 $β$ 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 $β$ 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. | |
| dc.description | 18 pages, 4 figures | |
| dc.identifier | https://arxiv.org/abs/0706.4082 | |
| dc.identifier | http://arxiv.org/abs/0706.4082 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/132555 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | Functional Analysis | |
| dc.subject | 76D03, 46E35, 42A16 | |
| dc.title | Inf-sup estimates for the Stokes problem in a periodic channel | |
| dc.type | text |