Abstract
Estimates are derived for the so-called indeterminacy set formed by solutions to an elliptic type boundary value problem with not fully determined coefficients. A two-sided estimate for the diameter of the indeterminacy set is obtained in the energy norm. It is shown that this estimate depends on parameters defining the variability range of the coefficients. The analysis is based on functional a posteriori estimates that provide guaranteed bounds of the difference between an approximate solution and any admissible function in the energy space. The estimates are obtained for the diffusion equation. However, the proposed tools can be used for other classes of partial differential equations if functional a posteriori estimates are established. Bibliography: 2 titles.
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References
P. Neittaanmäki and S. Repin, Reliable Methods for Computer Simulation, Error Control and a Posteriori Estimates, Elsevier, 2004.
S. Repin, “A posteriori error estimation for variational problems with uniformly convex functionals,” Math. Comput. 69 (2000), 481–500.
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Translated from Problemy Matematicheskogo Analiza, No. 36, 2007, pp. 77–80.
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Maly, O., Repin, S. Estimates of the indeterminacy set for elliptic boundary value problems with uncertain data. J Math Sci 150, 1869–1874 (2008). https://doi.org/10.1007/s10958-008-0101-5
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DOI: https://doi.org/10.1007/s10958-008-0101-5