Finite Element Analysis of a Quadratic Eigenvalue Problem Arising in Dissipative Acoustics

Abstract

A quadratic eigenvalue problem arising in the determination of the vibration modes of an acoustic fluid contained in a cavity with absorbing walls is considered. The problem is shown to be equivalent to the spectral problem for a noncompact operator and a thorough spectral characterization is given. A numerical discretization based on Raviart--Thomas finite elements is analyzed. The method is proved to be free of spurious modes and to converge with optimal order. Implementation issues and numerical experiments confirming the theoretical results are reported.