A priori hp-estimates for discontinuous Galerkin approximations to linear hyperbolic integro-differential equations

Samir Karaa, Amiya K. Pani, Sangita Yadav

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Abstract

An hp-discontinuous Galerkin (DG) method is applied to a class of second order linear hyperbolic integro-differential equations. Based on the analysis of an expanded mixed type Ritz-Volterra projection, a priori hp-error estimates in L∞(L2)-norm of the velocity as well as of the displacement, which are optimal in the discretizing parameter h and suboptimal in the degree of polynomial p are derived. For optimal estimates of the displacement in L∞(L2)-norm with reduced regularity on the exact solution, a variant of Baker's nonstandard energy formulation is developed and analyzed. Results on order of convergence which are similar in spirit to linear elliptic and parabolic problems are established for the semidiscrete case after suitably modifying the numerical fluxes. For the completely discrete scheme, an implicit-in-time procedure is formulated, stability results are derived and a priori error estimates are discussed. Finally, numerical experiments on two dimensional domains are conducted which confirm the theoretical results.

Original languageEnglish
Pages (from-to)1-23
Number of pages23
JournalApplied Numerical Mathematics
Volume96
DOIs
Publication statusPublished - May 6 2015

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Keywords

  • hp-Error estimates
  • Linear second order hyperbolic integro-differential equation
  • Local discontinuous Galerkin method
  • Mixed type Ritz-Volterra projection
  • Nonstandard formulation
  • Numerical experiments
  • Order of convergence
  • Role of stabilizing parameters
  • Semidiscrete and completely discrete schemes

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics
  • Numerical Analysis

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