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Mixed FEM for Time-Fractional Diffusion Problems with Time-Dependent Coefficients

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Abstract

In this paper, a mixed finite element method is applied in spatial directions while keeping time variable continuous to a class of time-fractional diffusion problems with time-dependent coefficients on a bounded convex polygonal domain. Based on an energy argument combined with a repeated application of an integral operator, optimal error estimates, which are optimal with respect to both approximation properties and regularity results, are derived for the semidiscrete problem with smooth as well as nonsmooth initial data. Specially, a priori error bounds for both primary and secondary variables in \(L^2\)-norm are established. Since the comparison between Fortin projection and the mixed Galerkin approximation of the secondary variable yields an improved rate of convergence, therefore, as a by-product, we derive \(L^p\)-estimates for the error in primary variable. Finally, some numerical experiments are conducted to confirm our theoretical findings.

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Acknowledgements

This research is supported by the Research Council of Oman grant ORG/CBS/15/001. The second author acknowledges the support from Institute Chair Professor’s fund and the support from SERB, Govt. India via MATRIX Grant No. MTR/201S/000309. Both the authors thank the referees for their valuable suggestions which help to improve the manuscript.

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Authors and Affiliations

  1. FracDiff Research Group, Department of Mathematics, Sultan Qaboos University, Al-Khod 123, Muscat, Oman

    Samir Karaa

  2. Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India

    Amiya K. Pani

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  1. Samir Karaa
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  2. Amiya K. Pani
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Correspondence to Amiya K. Pani.

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Karaa, S., Pani, A.K. Mixed FEM for Time-Fractional Diffusion Problems with Time-Dependent Coefficients. J Sci Comput 83, 51 (2020). https://doi.org/10.1007/s10915-020-01236-7

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  • DOI: https://doi.org/10.1007/s10915-020-01236-7

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