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Time-stepping discontinuous Galerkin methods for fractional diffusion problems

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Abstract

Time-stepping \(hp\)-versions discontinuous Galerkin (DG) methods for the numerical solution of fractional subdiffusion problems of order \(-\alpha \) with \(-1<\alpha <0\) will be proposed and analyzed. Generic \(hp\)-version error estimates are derived after proving the stability of the approximate solution. For \(h\)-version DG approximations on appropriate graded meshes near \(t=0\), we prove that the error is of order \(O(k^{\max \{2,p\}+\frac{\alpha }{2}})\), where \(k\) is the maximum time-step size and \(p\ge 1\) is the uniform degree of the DG solution. For \(hp\)-version DG approximations, by employing geometrically refined time-steps and linearly increasing approximation orders, exponential rates of convergence in the number of temporal degrees of freedom are shown. Finally, some numerical tests are given.

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Authors and Affiliations

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

    Kassem Mustapha

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  1. 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 material of the paragraph: Motivation of the \(hp\) -DG and future work, on page 3 is mainly based on my discussions with Professor Michael Griebel especially during my visit to Bonn on May 2014. 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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Mustapha, K. Time-stepping discontinuous Galerkin methods for fractional diffusion problems. Numer. Math. 130, 497–516 (2015). https://doi.org/10.1007/s00211-014-0669-2

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  • DOI: https://doi.org/10.1007/s00211-014-0669-2

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