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A hybridizable discontinuous Galerkin method for fractional diffusion problems

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Abstract

We study the use of the hybridizable discontinuous Galerkin (HDG) method for numerically solving fractional diffusion equations of order \(-\alpha \) with \(-1<\alpha <0\). For exact time-marching, we derive optimal algebraic error estimates assuming that the exact solution is sufficiently regular. Thus, if for each time \(t \in [0,T]\) the approximations are taken to be piecewise polynomials of degree \(k\ge 0\) on the spatial domain \(\varOmega \), the approximations to \(u\) in the \(L_\infty \bigr (0,T;L_2(\varOmega )\bigr )\)-norm and to \(\nabla u\) in the \(L_\infty \bigr (0,T;\mathbf{L}_2(\varOmega )\bigr )\)-norm are proven to converge with the rate \(h^{k+1}\), where \(h\) is the maximum diameter of the elements of the mesh. Moreover, for \(k\ge 1\) and quasi-uniform meshes, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for \(u\) converging with a rate of \(\sqrt{\log (T h^{-2/(\alpha +1)})}\, \,h^{k+2}\).

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Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, USA

    Bernardo Cockburn

  2. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals (KFUPM), Dhahran, Saudi Arabia

    Kassem Mustapha

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  1. Bernardo Cockburn
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  2. Kassem Mustapha
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Correspondence to Kassem Mustapha.

Additional information

The valuable comments of the editor and the referees improved the paper. The support of the Science Technology Unit at KFUPM through King Abdulaziz City for Science and Technology (KACST) under National Science, Technology and Innovation Plan (NSTIP) project No. 13-MAT1847-04 is gratefully acknowledged.

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Cockburn, B., Mustapha, K. A hybridizable discontinuous Galerkin method for fractional diffusion problems. Numer. Math. 130, 293–314 (2015). https://doi.org/10.1007/s00211-014-0661-x

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  • DOI: https://doi.org/10.1007/s00211-014-0661-x

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