Some error estimates for the finite volume element method for a parabolic problem

Panagiotis Chatzipantelidis; Raytcho Lazarov; Vidar Thomée

doi:10.1515/cmam-2012-0006

Some error estimates for the finite volume element method for a parabolic problem

Panagiotis Chatzipantelidis, Raytcho Lazarov, Vidar Thomée

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

13 Scopus citations

Abstract

We study spatially semidiscrete and fully discrete finite volume element methods for the homogeneous heat equation with homogeneous Dirichlet boundary conditions and derive error estimates for smooth and nonsmooth initial data. We show that the results of our earlier work [Math. Comp. 81 (2012), 1-20] for the lumped mass method carry over to the present situation. In particular, in order for error estimates for initial data only in L2 to be of optimal second order for positive time, a special condition is required, which is satisfied for symmetric triangulations. Without any such condition, only first order convergence can be shown, which is illustrated by a counterexample. Improvements hold for triangulations that are almost symmetric and piecewise almost symmetric. © 2013 Institute of Mathematics.

Original language	English (US)
Pages (from-to)	251-279
Number of pages	29
Journal	Computational Methods in Applied Mathematics
Volume	13
Issue number	3
DOIs	https://doi.org/10.1515/cmam-2012-0006
State	Published - Jul 1 2013
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2021-07-02
Acknowledged KAUST grant number(s): KUS-C1-016-04
Acknowledgements: The research of R. D. Lazarov was supported in parts by US NSF Grant DMS-1016525 and by award KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST). The research of P. Chatzipantelidis was partly supported by the FP7-REGPOT-2009-1 project “Archimedes Center for Modeling Analysis and Computation”, funded by the European Commission.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

Numerical Analysis
Computational Mathematics
Applied Mathematics

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title = "Some error estimates for the finite volume element method for a parabolic problem",

abstract = "We study spatially semidiscrete and fully discrete finite volume element methods for the homogeneous heat equation with homogeneous Dirichlet boundary conditions and derive error estimates for smooth and nonsmooth initial data. We show that the results of our earlier work [Math. Comp. 81 (2012), 1-20] for the lumped mass method carry over to the present situation. In particular, in order for error estimates for initial data only in L2 to be of optimal second order for positive time, a special condition is required, which is satisfied for symmetric triangulations. Without any such condition, only first order convergence can be shown, which is illustrated by a counterexample. Improvements hold for triangulations that are almost symmetric and piecewise almost symmetric. {\textcopyright} 2013 Institute of Mathematics.",

author = "Panagiotis Chatzipantelidis and Raytcho Lazarov and Vidar Thom{\'e}e",

note = "KAUST Repository Item: Exported on 2021-07-02 Acknowledged KAUST grant number(s): KUS-C1-016-04 Acknowledgements: The research of R. D. Lazarov was supported in parts by US NSF Grant DMS-1016525 and by award KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST). The research of P. Chatzipantelidis was partly supported by the FP7-REGPOT-2009-1 project “Archimedes Center for Modeling Analysis and Computation”, funded by the European Commission. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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doi = "10.1515/cmam-2012-0006",

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AU - Thomée, Vidar

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Y1 - 2013/7/1

N2 - We study spatially semidiscrete and fully discrete finite volume element methods for the homogeneous heat equation with homogeneous Dirichlet boundary conditions and derive error estimates for smooth and nonsmooth initial data. We show that the results of our earlier work [Math. Comp. 81 (2012), 1-20] for the lumped mass method carry over to the present situation. In particular, in order for error estimates for initial data only in L2 to be of optimal second order for positive time, a special condition is required, which is satisfied for symmetric triangulations. Without any such condition, only first order convergence can be shown, which is illustrated by a counterexample. Improvements hold for triangulations that are almost symmetric and piecewise almost symmetric. © 2013 Institute of Mathematics.

AB - We study spatially semidiscrete and fully discrete finite volume element methods for the homogeneous heat equation with homogeneous Dirichlet boundary conditions and derive error estimates for smooth and nonsmooth initial data. We show that the results of our earlier work [Math. Comp. 81 (2012), 1-20] for the lumped mass method carry over to the present situation. In particular, in order for error estimates for initial data only in L2 to be of optimal second order for positive time, a special condition is required, which is satisfied for symmetric triangulations. Without any such condition, only first order convergence can be shown, which is illustrated by a counterexample. Improvements hold for triangulations that are almost symmetric and piecewise almost symmetric. © 2013 Institute of Mathematics.

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JO - Computational Methods in Applied Mathematics

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Some error estimates for the finite volume element method for a parabolic problem

Abstract

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