A local discontinuous Galerkin method for the second-order wave equation

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Abstract

In this paper we present new superconvergence results for the local discontinuous Galerkin (LDG) method applied to the second-order scalar wave equation in one space dimension. Numerical experiments show O(hp+1)L2 convergence rate for the LDG solution and O(h p+2) superconvergent solutions at Radau points. More precisely, a local error analysis reveals that, at a fixed time t, the leading terms of the discretization errors for the solution and its derivative using p-degree polynomial approximations are proportional to the (p+1)-degree right Radau and (p+1)-degree left Radau polynomials, respectively. Thus, the p-degree LDG solution is O(h p+2) superconvergent at the roots of the (p+1)-degree right Radau polynomial and the derivative of the LDG solution is O(h p+2) superconvergent at the roots of the (p+1)-degree left Radau polynomial. These results are used to construct simple, efficient, and asymptotically correct a posteriori error estimates in regions where solutions are smooth. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori errors estimates under mesh refinement.

Original languageEnglish (US)
Pages (from-to)129-143
Number of pages15
JournalComputer Methods in Applied Mechanics and Engineering
Volume209-212
DOIs
Publication statusPublished - Feb 1 2012

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Keywords

  • A posteriori error estimation
  • Local discontinuous Galerkin method
  • Second-order wave equation
  • Superconvergence

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • Physics and Astronomy(all)
  • Computer Science Applications

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