Mixed approximation of the axisymmetric acoustic eigenvalue problem

Abstract

The aim of this paper is to study the numerical approximation of mixed formulations for the acoustic eigenvalue problem with axial symmetry. It is well known that, in the full three-dimensional problem, using Raviart–Thomas elements for the fluid displacements yields a discretization free of spurious eigenvalues. However, it has been recently shown that this is not the case in the axisymmetric pure displacement formulation. Two mixed formulations are introduced in order to avoid the spurious modes. A discretization based on Raviart–Thomas mixed method is proposed and analyzed. We provide a convergence result under rather general conditions, as well as absence of spurious modes. Moreover, we prove quasi-optimal order error estimates under additional regularity assumptions. Finally, we report numerical results which allow us to confirm the theoretical estimates.