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

Abstract

We develop a convergence theory of space-time discretizations for the linear, 2nd-order wave equation in polygonal domains $\Omega \subset {\mathbb{R}}^2$, possibly occupied by piecewise homogeneous media with different propagation speeds. Building on an unconditionally stable space-time DG formulation developed in [Moiola, Perugia 2018], 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 \emph{sparse} space-time version of the DG scheme. The latter scheme is based on the so-called \emph{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 $\Omega$ 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.

arXiv comment38 pages, 8 figures

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Cited by in corpus (1)

• A space-time quasi-Trefftz DG method for the wave equation with piecewise-smooth coefficients