Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems

Kassem Mustapha, Maher Nour, Bernardo Cockburn

Research output: Contribution to journalArticlepeer-review

25 Scopus citations

Abstract

We study the hybridizable discontinuous Galerkin (HDG) method for the spatial discretization of time fractional diffusion models with Caputo derivative of order 0 < α < 1. For each time t ∈ [0, T], when the HDG approximations are taken to be piecewise polynomials of degree k ≥ 0 on the spatial domain Ω, the approximations to the exact solution u in the L(0, T; L2(Ω))-norm and to ∇u in the (Formula presented.) -norm are proven to converge with the rate hk+1 provided that u is sufficiently regular, where h is the maximum diameter of the elements of the mesh. Moreover, for k ≥ 1, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for u converging with a rate hk+2 (ignoring the logarithmic factor), for quasi-uniform spatial meshes. Numerical experiments validating the theoretical results are displayed.

Original languageEnglish (US)
Pages (from-to)377-393
Number of pages17
JournalAdvances in Computational Mathematics
Volume42
Issue number2
DOIs
StatePublished - Apr 1 2016

Keywords

  • Anomalous diffusion
  • Convergence analysis
  • Discontinuous Galerkin methods
  • Hybridization
  • Time fractional

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