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A Note on a Posteriori Error Bounds for Numerical Solutions of Elliptic Equations with a Piecewise Constant Reaction Coefficient Having Large Jumps

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Abstract

We have derived guaranteed, robust, and fully computable a posteriori error bounds for approximate solutions of the equation \(\Delta \Delta u + {{\Bbbk }^{2}}u = f\), where the coefficient \(\Bbbk \geqslant 0\) is a constant in each subdomain (finite element) and chaotically varies between subdomains in a sufficiently wide range. For finite element solutions, these bounds are robust with respect to \(\Bbbk \in [0,{\text{c}}{{{\text{h}}}^{{ - 2}}}]\), \(c = {\text{const}}\), and possess some other good features. The coefficients in front of two typical norms on their right-hand sides are only insignificantly worse than those obtained earlier for \(\Bbbk \equiv {\text{const}}{\text{.}}\) The bounds can be calculated without resorting to the equilibration procedures, and they are sharp for at least low-order methods. The derivation technique used in this paper is similar to the one used in our preceding papers (2017–2019) for obtaining a posteriori error bounds that are not improvable in the order of accuracy.

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  1. St. Petersburg State University, 199034, St. Petersburg, Russia

    V. G. Korneev

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Korneev, V.G. A Note on a Posteriori Error Bounds for Numerical Solutions of Elliptic Equations with a Piecewise Constant Reaction Coefficient Having Large Jumps. Comput. Math. and Math. Phys. 60, 1754–1760 (2020). https://doi.org/10.1134/S096554252011007X

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