• Helmholtz equation;
  • plane wave basis;
  • evanescent wave basis;
  • ultra weak variational formulation;
  • upwind discontinuous Galerkin method;
  • element shapes


In this paper, we investigate strategies to improve the accuracy and efficiency of the ultra weak variational formulation (UWVF) of the Helmholtz equation. The UWVF is a Trefftz type, nonpolynomial method using basis functions derived from solutions of the adjoint Helmholtz equation. We shall consider three choices of basis function: propagating plane waves (original choice), Bessel basis functions, and evanescent wave basis functions. Traditionally, two-dimensional triangular elements are used to discretize the computational domain. However, the element shapes affect the conditioning of the UWVF. Hence, we investigate the use of different element shapes aiming to lower the condition number and number of degrees of freedom. Our results include the first tests of a plane wave method on meshes of mixed element types. In many modeling problems, evanescent waves occur naturally and are challenging to model. Therefore, we introduce evanescent wave basis functions for the first time in the UWVF to tackle rapidly decaying wave modes. The advantages of an evanescent wave basis are verified by numerical simulations on domains including curved interfaces.Copyright © 2013 John Wiley & Sons, Ltd.