Research reports

Years: 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

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

by P. Bansal and A. Moiola and I. Perugia and Ch. Schwab

(Report number 2020-10)

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~\cite{MoPe18}, 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.

Keywords: acoustic wave, point singularities, space-time DG, local isotropic mesh refinement, combination formula

@Techreport{BMPS20_883,
  author = {P. Bansal and A. Moiola and I. Perugia and Ch. Schwab},
  title = {Space–time discontinuous Galerkin approximation of acoustic waves with point singularities},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-10},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-10.pdf },
  year = {2020}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).