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Constructive error analysis of a full-discrete finite element method for the heat equation

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Abstract

We present a new full-discrete finite element method for the heat equation, and show the numerical stability of the method by verified computations. Since, in the error analysis, we use the constructive error estimates proposed by Nakao et al. (SIAM J Numer Anal 51(3):1525–1541, 2013) this work is considered as an extension of that paper. We emphasize that the concerned scheme seems to use the quite standard Galerkin method and is easy to implement for evolutionary equations compared with previous ones. In the constructive error estimates, we effectively use the numerical computations with guaranteed accuracy.

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References

  1. Kinoshita, T., Kimura, T., Nakao, M.T.: A posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations. J. Comput. Appl. Math. 236, 1622–1636 (2011)

    Article  MathSciNet  Google Scholar 

  2. Kinoshita, T., Kimura, T., Nakao, M.T.: On the a posteriori estimates for inverse operators of linear parabolic equations with applications to the numerical enclosure of solutions for nonlinear problems. Numer. Math. 126, 679–701 (2014)

    Article  MathSciNet  Google Scholar 

  3. Nakao, M.T.: Solving nonlinear parabolic problems with result verification. Part I: one-spacedimensional case. J. Comput. Appl. Math. 3, 323–334 (1991)

    Article  Google Scholar 

  4. Nakao, M.T., Yamamoto, N., Kimura, S.: On the best constant in the error bound for the \(H_0^1\)-projection into piecewise polynomial spaces. J. Approx. Theory 93, 491–500 (1998)

    Article  MathSciNet  Google Scholar 

  5. Nakao, M.T., Kimura, T., Kinoshita, T.: Constructive a priori error estimates for a full discrete approximation of the heat equation. SIAM J. Numer. Anal. 51(3), 1525–1541 (2013)

    Article  MathSciNet  Google Scholar 

  6. Rump, S.M.: INTLAB-INTerval LABoratory. In: Csendes, T., (ed.) Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, pp. 77–104 (1999). http://www.ti3.tu-harburg.de/rump/intlab/. Accessed 14 Dec 2017

  7. Schultz, M.H.: Spline Analysis. Prentice-Hall, Englewood Cliffs (1973)

    MATH  Google Scholar 

  8. Zeidler, E.: Nonlinear Functional Analysis and its Applications II/A. Springer, New York (1990)

    Book  Google Scholar 

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Acknowledgements

The authors are grateful to the referee for his/her very helpful comments and suggestion. This work was partially supported by JSPS KAKENHI Grant Number 17K17948, 18K03434 and 18K03440.

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Authors and Affiliations

  1. Division of Infant Education, Nakamura Gakuen Junior College, Fukuoka, 814-0198, Japan

    Kouji Hashimoto

  2. Department of Information Science, Saga University, Saga, 840-8502, Japan

    Takuma Kimura & Teruya Minamoto

  3. Faculty of Science and Engineering, Waseda University, Tokyo, 169-8555, Japan

    Mitsuhiro T. Nakao

Authors
  1. Kouji Hashimoto
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  2. Takuma Kimura
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  3. Teruya Minamoto
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  4. Mitsuhiro T. Nakao
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Correspondence to Kouji Hashimoto.

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Hashimoto, K., Kimura, T., Minamoto, T. et al. Constructive error analysis of a full-discrete finite element method for the heat equation. Japan J. Indust. Appl. Math. 36, 777–790 (2019). https://doi.org/10.1007/s13160-019-00362-6

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  • DOI: https://doi.org/10.1007/s13160-019-00362-6

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