Finite element approximation for an axisymmetric time-dependent acoustic problem

The aim of this paper is to study the numerical approximation of a mixed formulation for the time-domain axisymmetric acoustic problem. We show that non-physical oscillations appear when lowest order triangular Raviart–Thomas finite elements are used to discretize the problem. We analyze a weak formulation with the unknowns being the displacement and the acoustic potential, which allows us to avoid this drawback. For its numerical approximation, we first propose a semidiscrete in space discretization based on Raviart–Thomas mixed method. We derive error estimates in L∞(L2)-norms for the proposed scheme. Then, we consider a fully discrete approximation based on an implicit finite difference scheme in time, from which we obtain optimal error estimates for sufficiently smooth solutions. Finally, we report some numerical results that allow us to assess the performance of the method. These results also show that the numerical solution is not polluted by non-physical oscillations, as is the case with other alternative approaches.