Space–time discontinuous Galerkin approximation of acoustic waves with point singularities

Abstract

Abstract We develop a convergence theory of space–time discretizations for the linear, second-order wave equation in polygonal domains $\varOmega \subset{\mathbb R}^2$, possibly occupied by piecewise homogeneous media with different propagation speeds. Building on an unconditionally stable space–time DG formulation developed in Moiola & Perugia (2018, A space–time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation. Numer. Math., 138, 389–435), we (a) prove optimal convergence rates for the space–time scheme with local isotropic corner mesh refinement on the spatial domain, and (b) demonstrate numerically optimal convergence rates of a suitable sparse space–time version of the DG scheme. The latter scheme is based on the so-called combination formula, in conjunction with a family of anisotropic space–time DG discretizations. It results in optimal-order convergent schemes, also in domains with corners, with a number of degrees of freedom that scales essentially like the DG solution of one stationary elliptic problem in $\varOmega $ on the finest spatial grid. Numerical experiments for both smooth and singular solutions support convergence rate optimality on spatially refined meshes of the full and sparse space–time DG schemes.