Superconvergence of the local discontinuous galerkin method applied to the one-dimensional second-order wave equation

Research output: Contribution to journalArticle

15 Scopus citations

Abstract

We analyze the superconvergence properties of the local discontinuous Galerkin (LDG) method applied to the second-order wave equation in one space dimension. With a suitable projection of the initial conditions for the LDG scheme, we prove that the LDG solution and its spatial derivative are O (h p + 3 / 2) super close to particular projections of the exact solutions for pth-degree polynomial spaces. We use these results to show that the significant parts of the discretization errors for the LDG solution and its derivative are proportional to (p + 1) -degree right and left Radau polynomials, respectively. These results allow us to prove that the p-degree LDG solution and its derivative are O (h p + 3 / 2) superconvergent at the roots of (p + 1) -degree right and left Radau polynomials, respectively, while computational results show higher O (h p + 2) convergence rate. Superconvergence results can be used to construct asymptotically correct a posteriori error estimates by solving a local steady problem on each element. This will be discussed further in Part II of this work, where we will prove that the a posteriori LDG error estimates for the solution and its derivative converge to the true spatial errors in the L 2-norm under mesh refinement.

Original languageEnglish (US)
Pages (from-to)862-901
Number of pages40
JournalNumerical Methods for Partial Differential Equations
Volume30
Issue number3
DOIs
StatePublished - May 2014

Keywords

  • Radau points
  • alternating flux
  • error estimate
  • local discontinuous Galerkin method
  • projection
  • second-order wave equation
  • superconvergence

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

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