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 Ω ⊂ R 2 , possibly occupied by piecewise homogeneous media with different propagation speeds. Building on an unconditionally stable space-time DG formulation developed in [32], 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.

A discontinuous Galerkin time integration scheme for second order differential equations with applications to seismic wave propagation problems

“…In practice, since we use discontinuous functions, we can compute the numerical solution one time slab at time, assuming the initial conditions stemming from the previous time slab known. Hence, problem (14) reduces to: find z ∈ V rn (I n ) such that…”

“…In structured techniques [17,56], the space-time grid is the cartesian product of a spatial mesh and a time partition. Examples of applications to second order hyperbolic problems can be found in [55,29,14]. Unstructured techniques [37,38] employ grids generated considering the time as an additional dimension.…”