Space-time discontinuous Galerkin approximation of acoustic waves with point singularities
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We develop a convergence theory of space-time discretizations for the linear, 2nd-order wave equation in polygonal Ω⊂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 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, in work and memory which scales essentially (up to logarithmic terms) like the DG solution of one stationary elliptic problem in Ω 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.
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