We develop a convergence theory of space-time discretizations for the linear,
2nd-order wave equation in polygonal domains $Ømega\subsetR^2$,
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, with a number
of degrees of freedom that scales essentially like the DG solution of one
stationary elliptic problem in $Ømega$ 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.
Description
Space-time discontinuous Galerkin approximation of acoustic waves with point singularities - 2002.11575.pdf
%0 Generic
%1 bansal2020spacetime
%A Bansal, Pratyuksh
%A Moiola, Andrea
%A Perugia, Ilaria
%A Schwab, Christoph
%D 2020
%K convergence spacetime
%T Space-time discontinuous Galerkin approximation of acoustic waves with
point singularities
%U http://arxiv.org/abs/2002.11575
%X We develop a convergence theory of space-time discretizations for the linear,
2nd-order wave equation in polygonal domains $Ømega\subsetR^2$,
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, with a number
of degrees of freedom that scales essentially like the DG solution of one
stationary elliptic problem in $Ømega$ 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.
@misc{bansal2020spacetime,
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.},
added-at = {2021-06-09T16:53:11.000+0200},
author = {Bansal, Pratyuksh and Moiola, Andrea and Perugia, Ilaria and Schwab, Christoph},
biburl = {https://www.bibsonomy.org/bibtex/2990efcf000ccb6c982884c351f052493/dagon},
description = {Space-time discontinuous Galerkin approximation of acoustic waves with point singularities - 2002.11575.pdf},
interhash = {94254d8935053303c7cdee318e14854d},
intrahash = {990efcf000ccb6c982884c351f052493},
keywords = {convergence spacetime},
note = {cite arxiv:2002.11575Comment: 38 pages, 8 figures},
timestamp = {2021-06-09T16:53:35.000+0200},
title = {Space-time discontinuous Galerkin approximation of acoustic waves with
point singularities},
url = {http://arxiv.org/abs/2002.11575},
year = 2020
}