MATHEMATICS OF COMPUTATION Volume 71, Number 238, Pages 537–552 S 0025-5718(01)01335-7 Article electronically published on September 17, 2001 ANALYSIS OF A FINITE ELEMENT METHOD FOR PRESSURE/POTENTIAL FORMULATION OF ELASTOACOUSTIC SPECTRAL PROBLEMS ALFREDO BERM ´ UDEZ AND RODOLFO RODR ´ IGUEZ Abstract. A finite element method to approximate the vibration modes of a structure enclosing an acoustic fluid is analyzed. The fluid is described by using simultaneously pressure and displacement potential variables, whereas displacement variables are used for the solid. A mathematical analysis of the continuous spectral problem is given. The problem is discretized on a simplicial mesh by using piecewise constant elements for the pressure and continuous piecewise linear finite elements for the other fields. Error estimates are settled for approximate eigenvalues and eigenfrequencies. Finally, implementation issues are discussed. 1. Introduction In this paper we analyze a finite element method for the numerical solution of a spectral problem arising in fluid-solid interactions. It concerns the numerical computation of internal elastoacoustic vibrations, i.e., harmonic vibrations of a coupled system consisting of an elastic solid enclosing an acoustic (compressible, inviscid and barotropic) fluid. A first possibility to solve this problem is to consider a formulation in terms of displacements in the solid and pressure in the fluid (see [17]). However, such an approach leads to nonsymmetric eigenvalue problems, which is an inconvenient from the numerical point of view. An alternative procedure has been recently introduced in [5] (see also [2, 3, 4]). It is based on using displacement variables also for the fluid, discretized by lowest degree Raviart-Thomas finite elements on a triangular (or tetrahedral) mesh. Interface coupling between this discretization and classical piecewise linear finite elements for the solid displacements is achieved in a nonconforming way. Error estimates have been obtained and it has been proved that no spurious modes arise as is typical in other discretizations of this formulation (see [10]). Another approach, also leading to symmetrical spectral problems, has been intro- duced in [13]. It consists of using simultaneously the pressure and the potential of displacements to describe the fluid motion. In the present paper we analyze a finite Received by the editor April 13, 1999 and, in revised form, August 14, 2000. 2000 Mathematics Subject Classification. Primary 65N25, 65N30; Secondary 70J30, 74F10, 76Q05. Key words and phrases. Finite element spectral approximation, elastoacoustic vibrations. The first author was supported by DGESIC project PB97-0508 (Spain). The second author was supported by FONDECYT No. 1.990.346 and FONDAP in Applied Mathematics (Chile). c 2001 American Mathematical Society 537 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use