A uniform discrete inf-sup inequality for finite element hydro-elastic models

George  Avalos; Daniel Toundykov

doi:10.3934/eect.2016017

Article Contents

2016, Volume 5, Issue 4: 515-531. Doi: 10.3934/eect.2016017

This issue Previous Article Sharp regularity theory of second order hyperbolic equations with Neumann boundary control non-smooth in space Next Article Linearized hydro-elasticity: A numerical study

A uniform discrete inf-sup inequality for finite element hydro-elastic models

George Avalos^1, and
Daniel Toundykov^2,

1.
Department of Mathematics, University of Nebraska, Lincoln, Lincoln, NE, 68588
2.
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588

Received: May 31, 2016

Revised: July 31, 2016

Published: November 2016

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Abstract

A seminal result concerning finite element (FEM) approximations of the Stokes equation was the discrete inf-sup inequality that is uniform with respect to the mesh size parameter. This inequality leads to optimal error estimates for the FEM scheme. The original version pertains to the Stokes system with non-slip boundary condition on the entire boundary. On the other hand, in fluid-structure interaction problems, the interface dynamics between the fluid and the solid satisfies velocity and stress matching constraints. As a result, the pressure variable is no longer determined up to a constant and becomes subject to non-homogeneous Dirichlet conditions on the common interface. In this context, we establish a uniform discrete inf-sup estimate for a fluid-structure FEM implementation based on Taylor-Hood elements, and use this inequality to verify some stability and error estimates of this numerical scheme. An added benefit of this framework is that it does not require the Poisson-equation approach to solve for the pressure variable.

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Mathematics Subject Classification: Primary: 76M10, 65N12; Secondary: 74F10, 76D07, 74B05, 74S05.

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