A space-time Trefftz discontinuous Galerkin method for the acoustic wave   equation in first-order formulation
  We introduce a space-time Trefftz discontinuous Galerkin method for the first-order transient acoustic wave equations in arbitrary space dimensions, extending the one dimensional scheme of Kretzschmar et al. (2016, IMA J. Numer. Anal., 36, 1599-1635). Test and trial discrete functions are space-time piecewise polynomial solutions of the wave equations. We prove well-posedness and a priori error bounds in both skeleton-based and mesh-independent norms. The space-time formulation corresponds to an implicit time-stepping scheme, if posed on meshes partitioned in time slabs, or to an explicit scheme, if posed on "tent-pitched" meshes. We describe two Trefftz polynomial discrete spaces, introduce bases for them and prove optimal, high-order $h$-convergence bounds. 
Andrea Moiola, Ilaria Perugia
math.NA