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A posteriori error estimates for approximations of evolutionary convection–diffusion problems

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We derive computable upper bounds for the difference between an exact solution of the evolutionary convection-diffusion problem and an approximation of this solution. The estimates are obtained by certain transformations of the integral identity that defines the generalized solution. These estimates depend on neither special properties of the exact solution nor its approximation and involve only global constants coming from embedding inequalities. The estimates are first derived for functions in the corresponding energy space, and then possible extensions to classes of piecewise continuous approximations are discussed. Bibliography: 7 titles.

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References

  1. O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics [in Russian], Nauka, Moscow (1973); English transl.: Springer, New York (1985).

    Google Scholar 

  2. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967); English transl.: Am. Math. Soc., Providence, RI (1968).

    Google Scholar 

  3. S. Repin, A Posteriori Estimates for Partial Differential Equations, Walter de Gruyter, Berlin (2008).

    Book  MATH  Google Scholar 

  4. S. Repin, “Estimates of deviation from exact solutions of initial–boundary value problems for the heat equation,” Rend. Mat. Acc. Lincei 13, 121–133 (2002).

    MATH  MathSciNet  Google Scholar 

  5. A. V. Gaevskaya and S. I. Repin, “A posteriori error estimates for approximate solutions of linear parabolic problems” [in Russian], Differ. Uravn. 41, No. 7, 925–937 (2005); English transl.: Differ. Equ. 41, No. 7, 970–983 (2005).

    MathSciNet  Google Scholar 

  6. P. Neittanamäki and S. Repin, “A posteriori error majorants for approximations of the evolutionary Stokes problem,” J. Numer. Math. 18, No. 2, 119–134 (2010).

    Article  MathSciNet  Google Scholar 

  7. R. Lazarov, S. Repin, and S. Tomar, “Functional a posteriori error estimates for discontinuous Galerkin method,” Numer. Methods Partial Differ. Equ. 25, 952–971 (2009).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. V. A. Steklov Institute of Mathematics, Russian Academy of Sciences, 27, Fontanka, St. Petersburg, 191023, Russia

    S. I. Repin

  2. Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, 69, Altenbergerstr., Linz, 4040, Austria

    S. K. Tomar

Authors
  1. S. I. Repin
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  2. S. K. Tomar
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Correspondence to S. I. Repin.

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Translated from Problems in Mathematical Analysis 50, September 2010, pp. 113–123

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Repin, S.I., Tomar, S.K. A posteriori error estimates for approximations of evolutionary convection–diffusion problems. J Math Sci 170, 554–566 (2010). https://doi.org/10.1007/s10958-010-0100-1

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  • DOI: https://doi.org/10.1007/s10958-010-0100-1

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