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

David Bourne; Howard Elman; John E. Osborn

doi:10.3934/cpaa.2009.8.143

Article Contents

2009, Volume 8, Issue 1: 143-160. Doi: 10.3934/cpaa.2009.8.143

This issue Previous Article A Non-Self-Adjoint Quadratic Eigenvalue Problem Describing a Fluid-Solid Interaction Part I: Formulation, Analysis, and Computations Next Article A Numerical Method for a Non-Smooth Advection-Diffusion Problem Arising in Sand Mechanics

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

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
3.
Department of Mathematics, University of Maryland, College Park, MD 20742

Received: May 31, 2008

Revised: August 31, 2008

Published: December 2008

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Abstract

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,
fluid-solid interaction,
viscous fluid,
nonlinear viscoeastic shell.

Mathematics Subject Classification: 65N25, 74F10, 76D05, 74D10, 65N12.

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

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