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

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We derive estimates of the Babuska-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 ... continued below

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Wilkening, Jon December 10, 2008.

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We derive estimates of the Babuska-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{sup 1,1} or even C{sup 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{sup 1,1} case, and nearly optimal in the Lipschitz case.

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  • Journal Name: SIAM Journal on Mathematical Analysis

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  • Report No.: LBNL-1295E
  • Grant Number: DE-AC02-05CH11231
  • Office of Scientific & Technical Information Report Number: 944571
  • Archival Resource Key: ark:/67531/metadc900046

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  • December 10, 2008

Added to The UNT Digital Library

  • Sept. 27, 2016, 1:39 a.m.

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  • Nov. 8, 2016, 1:10 p.m.

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Wilkening, Jon. Inf-sup estimates for the Stokes problem in a periodic channel, article, December 10, 2008; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc900046/: accessed November 23, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.