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

Deepjyoti Goswami; Amiya K. Pani; Sangita Yadav

doi:10.1007/s10915-012-9666-8

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

Deepjyoti Goswami, Amiya K. Pani, Sangita Yadav

Research output: Contribution to journal › Article › peer-review

18 Scopus citations

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 L2 L2-error estimates are derived for semidiscrete approximations, when the initial condition is in L2 L2. 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 L2, L 2, which improves upon the results available in the literature. © 2013 Springer Science+Business Media New York.

Original language	English (US)
Pages (from-to)	131-164
Number of pages	34
Journal	Journal of Scientific Computing
Volume	56
Issue number	1
DOIs	https://doi.org/10.1007/s10915-012-9666-8
State	Published - May 1 2013

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-C1-013-04
Acknowledgements: 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.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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

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 L2 L2-error estimates are derived for semidiscrete approximations, when the initial condition is in L2 L2. 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 L2, L 2, which improves upon the results available in the literature. {\textcopyright} 2013 Springer Science+Business Media New York.",

author = "Deepjyoti Goswami and Pani, {Amiya K.} and Sangita Yadav",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KUK-C1-013-04 Acknowledgements: 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. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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AU - Pani, Amiya K.

AU - Yadav, Sangita

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KUK-C1-013-04 Acknowledgements: 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. This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2013/5/1

Y1 - 2013/5/1

N2 - 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 L2 L2-error estimates are derived for semidiscrete approximations, when the initial condition is in L2 L2. 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 L2, L 2, which improves upon the results available in the literature. © 2013 Springer Science+Business Media New York.

AB - 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 L2 L2-error estimates are derived for semidiscrete approximations, when the initial condition is in L2 L2. 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 L2, L 2, which improves upon the results available in the literature. © 2013 Springer Science+Business Media New York.

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JF - Journal of Scientific Computing

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ER -

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

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