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

Handle URI:
http://hdl.handle.net/10754/599091
Title:
Optimal Error Estimates of Two Mixed Finite Element Methods for Parabolic Integro-Differential Equations with Nonsmooth Initial Data
Authors:
Goswami, Deepjyoti; Pani, Amiya K.; Yadav, Sangita
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.
Citation:
Goswami D, Pani AK, Yadav S (2013) Optimal Error Estimates of Two Mixed Finite Element Methods for Parabolic Integro-Differential Equations with Nonsmooth Initial Data. Journal of Scientific Computing 56: 131–164. Available: http://dx.doi.org/10.1007/s10915-012-9666-8.
Publisher:
Springer Nature
Journal:
Journal of Scientific Computing
KAUST Grant Number:
KUK-C1-013-04
Issue Date:
1-May-2013
DOI:
10.1007/s10915-012-9666-8
Type:
Article
ISSN:
0885-7474; 1573-7691
Sponsors:
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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Full metadata record

DC FieldValue Language
dc.contributor.authorGoswami, Deepjyotien
dc.contributor.authorPani, Amiya K.en
dc.contributor.authorYadav, Sangitaen
dc.date.accessioned2016-02-25T13:52:42Zen
dc.date.available2016-02-25T13:52:42Zen
dc.date.issued2013-05-01en
dc.identifier.citationGoswami D, Pani AK, Yadav S (2013) Optimal Error Estimates of Two Mixed Finite Element Methods for Parabolic Integro-Differential Equations with Nonsmooth Initial Data. Journal of Scientific Computing 56: 131–164. Available: http://dx.doi.org/10.1007/s10915-012-9666-8.en
dc.identifier.issn0885-7474en
dc.identifier.issn1573-7691en
dc.identifier.doi10.1007/s10915-012-9666-8en
dc.identifier.urihttp://hdl.handle.net/10754/599091en
dc.description.abstractIn 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.en
dc.description.sponsorshipThe 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.en
dc.publisherSpringer Natureen
dc.subjectNonsmooth initial dataen
dc.subjectOptimal error estimatesen
dc.subjectParabolic integro-differential equationen
dc.subjectSemidiscrete Galerkin approximationen
dc.subjectTwo mixed finite element methodsen
dc.titleOptimal Error Estimates of Two Mixed Finite Element Methods for Parabolic Integro-Differential Equations with Nonsmooth Initial Dataen
dc.typeArticleen
dc.identifier.journalJournal of Scientific Computingen
dc.contributor.institutionUniversidade Tecnologica Federal do Parana, Curitiba, Brazilen
dc.contributor.institutionIndian Institute of Technology, Bombay, Mumbai, Indiaen
dc.contributor.institutionBirla Institute of Technology and Science Pilani, Pilani, Indiaen
kaust.grant.numberKUK-C1-013-04en
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