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

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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; L 2(Ω))-norm and to ∇u in the \(L_{\infty }(0, \textit {T}; \mathbf {L}_{2}({\Omega }))\)-norm are proven to converge with the rate h k+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 h k+2 (ignoring the logarithmic factor), for quasi-uniform spatial meshes. Numerical experiments validating the theoretical results are displayed.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    Kassem Mustapha & Maher Nour

  2. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Bernardo Cockburn

Authors
  1. Kassem Mustapha
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  2. Maher Nour
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  3. Bernardo Cockburn
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Correspondence to Kassem Mustapha.

Additional information

Communicated by: Jan Hesthaven

Support of the King Fahd University of Petroleum and Minerals (KFUPM) through the project FT131011 is gratefully acknowledged.

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Mustapha, K., Nour, M. & Cockburn, B. Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems. Adv Comput Math 42, 377–393 (2016). https://doi.org/10.1007/s10444-015-9428-x

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  • DOI: https://doi.org/10.1007/s10444-015-9428-x

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