L.L.Thompson: Finite element methods for acoustics, Preprint: J.Acoust.Soc.Am. A review of ¯nite element methods for time-harmonic acoustics Lonny L. Thompson Department of Mechanical Engineering, Clemson University Clemson, South Carolina, 29634-0921, USA Email:
[email protected] (Dated: Submitted July 8, 2004; Resubmitted May 28, 2005; Revised Dec 10, 2005; Accepted Dec 12, 2005) PACS numbers: 43.20.Bi,43.20.Fn,43.20.Rz Keywords: acoustics; Helmholtz equation; ¯nite element method; stabilized methods; unbounded domains; absorbing boundary condition; nonreflecting boundary condition; absorbing layer; in¯nite element; complex- symmetric; domain decomposition method; adaptive method; a posteriori error estimate; inverse scattering; optimization 1 Abstract State-of-the-art ¯nite element methods for time-harmonic acoustics governed by the Helmholtz equation are reviewed. Four major current challenges in the ¯eld are speci¯cally addressed: the e®ective treatment of acoustic scattering in unbounded domains, including local and nonlocal absorbing boundary conditions, in¯nite elements, and absorbing layers; numerical dispersion errors that arise in the approximation of short unresolved waves, pol- luting resolved scales, and requiring a large computational e®ort; e±cient algebraic equation solving methods for the resulting complex-symmetric (non-Hermitian) matrix systems in- cluding sparse iterative and domain decomposition methods; and a posteriori error estimates for the Helmholtz operator required for adaptive methods. Mesh resolution to control phase error and bound dispersion or pollution errors measured in global norms for large wave numbers in ¯nite element methods are described. Stabilized, multiscale and other wave- based discretization methods developed to reduce this error are reviewed.