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Journal of Mathematical Sciences

, Volume 170, Issue 4, pp 554–566 | Cite as

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

  • S. I. Repin
  • S. K. Tomar
Article

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.

Keywords

Radon Steklov Institute Integral Identity Posteriori Error Estimate Young Inequality 
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References

  1. 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. 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. 3.
    S. Repin, A Posteriori Estimates for Partial Differential Equations, Walter de Gruyter, Berlin (2008).MATHCrossRefGoogle Scholar
  4. 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).MATHMathSciNetGoogle Scholar
  5. 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).MathSciNetGoogle Scholar
  6. 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).CrossRefMathSciNetGoogle Scholar
  7. 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).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  1. 1.V. A. Steklov Institute of MathematicsRussian Academy of SciencesSt. PetersburgRussia
  2. 2.Johann Radon Institute for Computational and Applied MathematicsAustrian Academy of SciencesLinzAustria

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