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MiS Preprint
56/2008

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

David Bourne, Howard Elman and John Osborn

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.

Received:
Sep 3, 2008
Published:
Sep 3, 2008
MSC Codes:
65N25, 74F10, 76D05, 74D10, 65N12
Keywords:
Non-self-adjoint quadratic eigenvalue problem, fluid-solid interaction, viscous fluid, nonlinear viscoeastic shell

Related publications

inJournal
2009 Repository Open Access
David Bourne, Howard Elman and John E. Osborn

A non-self-adjoint quadratic eigenvalue problem describing a fluid-solid interaction. Pt. 2 : Analysis of convergence

In: Communications on pure and applied analysis, 8 (2009) 1, pp. 143-160