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Optimal Error Estimates of Two Mixed Finite Element Methods for Parabolic Integro-Differential Equations with Nonsmooth Initial Data

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Abstract

In the first part of this article, a new mixed method is proposed and analyzed for parabolic integro-differential equations (PIDE) with nonsmooth initial data. Compared to the standard mixed method for PIDE, the present method does not bank on a reformulation using a resolvent operator. Based on energy arguments combined with a repeated use of an integral operator and without using parabolic type duality technique, optimal \(L^2\)-error estimates are derived for semidiscrete approximations, when the initial condition is in \(L^2\). Due to the presence of the integral term, it is, further, observed that a negative norm estimate plays a crucial role in our error analysis. Moreover, the proposed analysis follows the spirit of the proof techniques used in deriving optimal error estimates for finite element approximations to PIDE with smooth data and therefore, it unifies both the theories, i.e., one for smooth data and other for nonsmooth data. Finally, we extend the proposed analysis to the standard mixed method for PIDE with rough initial data and provide an optimal error estimate in \(L^2,\) which improves upon the results available in the literature.

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Acknowledgments

The first author would like to thank CSIR, Government of India for the financial support. The second author acknowledges the research support of the Department of Science and Technology, Government of India under DST-CNPq Indo-Brazil Project No. DST/INT/Brazil /RPO-05/2007 ( Grant No. 490795/2007-2). This publication is also based on the work (AKP) supported in part by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). The authors also thank the referees for their valuable suggestions.

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Authors and Affiliations

  1. Departamento de Matemática, UFPR, Centro Politécnico, Jardim das Américas, Caixa Postal 19081, CEP: 81531-980,  Curitiba, Paran, Brazil

    Deepjyoti Goswami

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

    Amiya K. Pani

  3. Department of Mathematics, Birla Institute of Technology and Science, Pilani Pilani,  333031, Rajasthan, India

    Sangita Yadav

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  1. Deepjyoti Goswami
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  2. Amiya K. Pani
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  3. Sangita Yadav
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Correspondence to Amiya K. Pani.

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Goswami, D., Pani, A.K. & Yadav, S. Optimal Error Estimates of Two Mixed Finite Element Methods for Parabolic Integro-Differential Equations with Nonsmooth Initial Data. J Sci Comput 56, 131–164 (2013). https://doi.org/10.1007/s10915-012-9666-8

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  • DOI: https://doi.org/10.1007/s10915-012-9666-8

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