INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2011; 85:380–402 Published online 8 July 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.2979 A comparison of wave-based discontinuous Galerkin, ultra-weak and least-square methods for wave problems G. Gabard 1, ∗, † , P. Gamallo 2 and T. Huttunen 3,4 1 Institute of Sound and Vibration Research, University of Southampton, SO17 1BJ, U.K. 2 Departamento de Matemática Aplicada II, Universidade de Vigo, Vigo 36310, Spain 3 Department of Physics, University of Kuopio, P.O. Box 1627, FI-70211 Kuopio, Finland 4 Kuava Ltd., Microkatu 1, 7 FI-70211 Kuopio, Finland SUMMARY Several numerical methods using non-polynomial interpolation have been proposed for wave propagation problems at high frequencies. The common feature of these methods is that in each element, the solution is approximated by a set of local solutions. They can provide very accurate solutions with a much smaller number of degrees of freedom compared to polynomial interpolation. There are however significant differences in the way the matching conditions enforcing the continuity of the solution between elements can be formulated. The similarities and discrepancies between several non-polynomial numerical methods are discussed in the context of the Helmholtz equation. The present comparison is concerned with the ultra- weak variational formulation (UWVF), the least-squares method (LSM) and the discontinuous Galerkin method with numerical flux (DGM). An analysis in terms of Trefftz methods provides an interesting insight into the properties of these methods. Second, the UWVF and the LSM are reformulated in a similar fashion to that of the DGM. This offers a unified framework to understand the properties of several non-polynomial methods. Numerical results are also presented to put in perspective the relative accuracy of the methods. The numerical accuracies of the methods are compared with the interpolation errors of the wave bases. Copyright  2010 John Wiley & Sons, Ltd. Received 11 July 2009; Revised 1 June 2010; Accepted 2 June 2010 KEY WORDS: Helmholtz; discontinuous Galerkin; ultra-weak; least square 1. INTRODUCTION Numerical methods based on non-polynomial interpolation are now recognized as an efficient way to circumvent the limitations of standard frequency-domain numerical methods for high frequencies. The rationale behind the wave-based methods is to exploit a priori information on the physics of the problem so as to devise efficient numerical schemes. For wave propagation problems, the key is to capture the dispersion properties of the waves (such as the wave number). In this way the dispersion error inherent to the numerical approximation is minimized and the pollution error can be significantly reduced. Accurate solutions can be obtained at a minute cost compared to more traditional numerical methods such as finite element methods using polynomial shape functions. A great variety of methods have been proposed. With the partition of unity method (PUM), the standard finite element polynomial basis is enriched by multiplication with a set of plane waves at each node [1–3]. For the discontinuous enrichment method (DEM), the standard finite element ∗ Correspondence to: G. Gabard, Institute of Sound and Vibration Research, University of Southampton, SO17 1BJ, U.K. † E-mail: gabard@soton.ac.uk Copyright  2010 John Wiley & Sons, Ltd.