A comparison of wave-based discontinuous Galerkin, ultra-weak and least-square methods for wave problems

Abstract

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.

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