Two-Sided Estimates of the Solution Set for the Reaction–Diffusion Problem with Uncertain Data

  • Olli MaliEmail author
  • Sergey Repin
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 15)


We consider linear reaction–diffusion problems with mixed Dirichlét–Neumann–Robin conditions. The diffusion matrix, reaction coefficient, and the coefficient in the Robin boundary condition are defined with an uncertainty which allow bounded variations around some given mean values. A solution to such a problem cannot be exactly determined (it is a function in the set of “possible solutions” formed by generalized solutions related to possible data). The problem is to find parameters of this set. In this paper, we show that computable lower and upper bounds of the diameter (or radius) of the set can be expressed throughout problem data and parameters that regulate the indeterminacy range. Our method is based on using a posteriori error majorants and minorants of the functional type (see [5, 6]), which explicitly depend on the coefficients and allow to obtain the corresponding lower and upper bounds by solving the respective extremal problems generated by indeterminacy of coefficients.


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© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of Mathematical Information TechnologyUniversity of JyväskyläJyväskyläFinland
  2. 2.V. A. Steklov Institute of Mathematics in St. PetersburgSt. PetersburgRussia

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