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On Positivity Preservation in Some Finite Element Methods for the Heat Equation

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Numerical Methods and Applications (NMA 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8962))

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Abstract

We consider the initial boundary value problem for the homogeneous heat equation, with homogeneous Dirichlet boundary conditions. By the maximum principle the solution is nonnegative for positive time if the initial data are nonnegative. We study to what extent this property carries over to some piecewise linear finite element discretizations, namely the Standard Galerkin method, the Lumped Mass method, and the Finite Volume Element method. We address both spatially semidiscrete and fully discrete methods.

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References

  1. Chatzipantelidis, P., Lazarov, R.D., Thomée, V.: Some error estimates for the lumped mass finite element method for a parabolic problem. Math. Comp. 81, 1–20 (2012)

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  2. Chatzipantelidis, P., Lazarov, R.D., Thomée, V.: Some error estimates for the finite volume element method for a parabolic problem. Comput. Methods Appl. Math. 13, 251–279 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chou, S.H., Li, Q.: Error estimates in \(L^2\), \(H^1\) and \(L^\infty \) in covolume methods for elliptic and parabolic problems: a unified approach. Math. Comp. 69, 103–120 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Drǎgǎnescu, A., Dupont, T.F., Scott, L.R.: Failure of the discrete maximum principle for an elliptic finite element problem. Math. Comp. 74, 1–23 (2004)

    Article  Google Scholar 

  5. Fujii, H.: Some remarks on finite element analysis of time-dependent field problems. In: Theory and Practice in Finite Element Structural Analysis, pp. 91–106. University of Tokyo Press, Tokyo (1973)

    Google Scholar 

  6. Schatz, A.H., Thomée, V., Wahlbin, L.B.: On positivity and maximum-norm contractivity in time stepping methods for parabolic equations. Comput. Methods Appl. Math. 10, 421–443 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems, 2nd edn. Springer, Heidelberg (2006)

    MATH  Google Scholar 

  8. Thomée, V., Wahlbin, L.B.: On the existence of maximum principles in parabolic finite element equations. Math. Comp. 77, 11–19 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, 412 96, Gothenburg, Sweden

    V. Thomée

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  1. V. Thomée
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Correspondence to V. Thomée .

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Editors and Affiliations

  1. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Dimov

  2. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Stefka Fidanova

  3. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Lirkov

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Thomée, V. (2015). On Positivity Preservation in Some Finite Element Methods for the Heat Equation. In: Dimov, I., Fidanova, S., Lirkov, I. (eds) Numerical Methods and Applications. NMA 2014. Lecture Notes in Computer Science(), vol 8962. Springer, Cham. https://doi.org/10.1007/978-3-319-15585-2_2

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  • DOI: https://doi.org/10.1007/978-3-319-15585-2_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-15584-5

  • Online ISBN: 978-3-319-15585-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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