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 Zh. Vychisl. Mat. Mat. Fiz., 2020, Volume 60, Number 11, Pages 1815–1822 (Mi zvmmf11154)

General numerical methods

A note on a posteriori error bounds for numerical solutions of elliptic equations with a piecewise constant reaction coefficient having large jumps

V. G. Korneev

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

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 $\Bbbk \in [0,{c}{{{h}}^{{ - 2}}}]$, $c={const}$ , 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 $\Bbbk \equiv {const}{.}$ 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.

Key words: a posteriori error bounds, singularly perturbed fourth-order elliptic equations, piecewise constant reaction coefficient, finite element method, sharp bounds.

DOI: https://doi.org/10.31857/S0044466920110071

English version:
Computational Mathematics and Mathematical Physics, 2020, 60:11, 1754–1760

Bibliographic databases:

UDC: 519.632.4
Revised: 28.05.2020
Accepted:07.07.2020

Citation: V. G. Korneev, “A note on a posteriori error bounds for numerical solutions of elliptic equations with a piecewise constant reaction coefficient having large jumps”, Zh. Vychisl. Mat. Mat. Fiz., 60:11 (2020), 1815–1822; Comput. Math. Math. Phys., 60:11 (2020), 1754–1760

Citation in format AMSBIB
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