INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2012; 89:403–417 Published online 26 July 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.3239 Overview of the discontinuous enrichment method, the ultra-weak variational formulation, and the partition of unity method for acoustic scattering in the medium frequency regime and performance comparisons Dalei Wang 3, * ,† , Radek Tezaur 1 , Jari Toivanen 1 and Charbel Farhat 1,2,3 1 Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305, U.S.A. 2 Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, U.S.A. 3 Institute for Computational & Mathematical Engineering, Stanford University, Stanford, CA 94305, U.S.A. SUMMARY The efficient finite element discretization of the Helmholtz equation becomes challenging in the medium frequency regime because of numerical dispersion, or what is often referred to in the literature as the pollu- tion effect. A number of FEMs with plane wave basis functions have been proposed to alleviate this effect, and improve on the unsatisfactory preasymptotic convergence of the polynomial FEM. These include the partition of unity method, the ultra-weak variational formulation, and the discontinuous enrichment method. A previous comparative study of the performance of such methods focused on the first two aforementioned methods only. By contrast, this paper provides an overview of all three methods and compares several aspects of their performance for an acoustic scattering benchmark problem in the medium frequency regime. It is found that the discontinuous enrichment method outperforms both the partition of unity method and the ultra-weak variational formulation by a significant margin. Copyright © 2011 John Wiley & Sons, Ltd. Received 21 March 2011; Revised 3 May 2011; Accepted 8 May 2011 KEY WORDS: Discontinuous enrichment method; Helmholtz; medium frequency regime; partition of unity method; ultra-weak variational formulation 1. INTRODUCTION The Helmholtz equation represents the time harmonic form of a PDE modeling the propagation of a wave. It has many applications ranging from acoustics, electromagnetics, to aerodynamics and quantum mechanics. It is well known that the numerical solution of this equation is particularly challenging. Traditional discretization methods such as the standard Galerkin FEM are well suited for elliptic boundary value problems. However, the Helmholtz operator becomes indefinite with an increasing wavenumber. The oscillatory character of the exact solution and an insufficient mesh res- olution combine to make the discrete solution dispersive, that is, with a discrete wavenumber that differs from that of the exact solution. This is a global phenomenon by which the error is introduced in the entire computational domain. It is commonly known as the pollution effect. For example, the global relative error of the FEM approximation with linear elements of the solution scales as O  .h/ 2  , where h denotes the element size [1]. Thus, as the wavenumber  is increased, reducing h at the same rate to keep the product h constant is not sufficient to maintain the error constant. For this reason, the standard Galerkin FEM (or the polynomial FEM) can become cost prohibitive *Correspondence to: Dalei Wang, Institute for Computational & Mathematical Engineering, Stanford University, Stanford, CA 94305, U.S.A. † E-mail: daleiw@stanford.edu Copyright © 2011 John Wiley & Sons, Ltd.