Plane Wave Discontinuous Galerkin Methods

Standard low order Lagrangian finite element discretization of boundary value problems for the Helmholtz equation −∆u−ωu = f are afflicted with the so-called pollution phenomenon: though for sufficiently small hω an accurate approximation of u is possible, the Galerkin procedure fails to provide it. Attempts to remedy this have focused on incorporating extra information in the form of plane wave functions x 7→ exp(iωd · x), |d| = 1, into the trial spaces. Prominent examples of such methods are the plane wave partition of unity finite element method of Babuska and Melenk [1], and the ultra-weak Galerkin discretization due to Cessenat and Despres [3, 4]. Both perform well in computations, see the articles by Monk and Hutunen [7–9] for computational results for the ultra-weak approach.