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    A Non-Self-Adjoint Quadratic Eigenvalue Problem Describing a Fluid-Solid Interaction Part I: Formulation, Analysis, and Computations
January  2009, 8(1): 143-160. doi: 10.3934/cpaa.2009.8.143

A Non-Self-Adjoint Quadratic Eigenvalue Problem Describing a Fluid-Solid Interaction Part II: Analysis of Convergence

David Bourne 1, , Howard Elman 2, and  John E. Osborn 3,

1. 

Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig
2. 

Department of Computer Science, Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742, United States
3. 

Department of Mathematics, University of Maryland, College Park, MD 20742, United States

Received  June 2008 Revised  September 2008 Published  October 2008

This paper is the second part of a two-part paper treating a non-self-adjoint quadratic eigenvalue problem for the linear stability of solutions to the Taylor-Couette problem for flow of a viscous liquid in a deformable cylinder, with the cylinder modelled as a membrane. The first part formulated the problem, analyzed it, and presented computations. In this second part, we first give a weak formulation of the problem, carefully contrived so that the pressure boundary terms are eliminated from the equations. We prove that the bilinear forms appearing in the weak formulation satisfy continuous inf-sup conditions. We combine a Fourier expansion with the finite element method to produce a discrete problem satisfying discrete inf-sup conditions. Finally, the Galerkin approximation theory for polynomial eigenvalue problems is applied to prove convergence of the spectrum.
Keywords: Non-self-adjoint quadratic eigenvalue problem, viscous fluid, nonlinear viscoeastic shell., fluid-solid interaction.
Mathematics Subject Classification: 65N25, 74F10, 76D05, 74D10, 65N1.
Citation: David Bourne, Howard Elman, John E. Osborn. A Non-Self-Adjoint Quadratic Eigenvalue Problem Describing a Fluid-Solid Interaction Part II: Analysis of Convergence. Communications on Pure & Applied Analysis, 2009, 8 (1) : 143-160. doi: 10.3934/cpaa.2009.8.143
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