Numerical analysis of a time domain elastoacoustic problem

This paper deals with the numerical analysis of a system of second order in time partial differential equations modeling the vibrations of a coupled system that consists of an elastic solid in contact with an inviscid compressible fluid. We analyze a weak formulation with the unknowns in both media being the respective displacement fields. For its numerical approximation, we propose first a semidiscrete in space discretization based on standard Lagrangian elements in the solid and Raviart–Thomas elements in the fluid. We establish its well-posedness and derive error estimates in appropriate norms for the proposed scheme. In particular, we obtain an $\mathrm L^{\infty }(\mathrm L^2)$ optimal rate of convergence under minimal regularity assumptions of the solution, which are proved to hold for appropriate data of the problem. Then we consider a fully discrete approximation based on a family of implicit finite difference schemes in time, from which we obtain optimal error estimates for sufficiently smooth solutions. Finally, we report some numerical results, which allow us to assess the performance of the method. These results also show that the numerical solution is not polluted by spurious modes as is the case with other alternative approaches.