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Zh. Vychisl. Mat. Mat. Fiz., 2020, Volume 60, Number 5, Pages 767–783 (Mi zvmmf11073)  

Estimates of the deviation from exact solutions of boundary value problems in measures stronger than the energy norm

S. I. Repin

Steklov Institute of Mathematics (St. Petersburg Branch), Russian Academy of Sciences, St. Petersburg, 191023 Russia

Abstract: The paper is concerned with estimates of the difference between a given function and the exact solution of an elliptic boundary value problem. Estimates of this type have been derived earlier in terms of the natural energy norm. In this work, an approach is proposed to obtain stronger measures of the deviation and relevant estimates applicable if the exact solution and the approximation have additional regularity (with respect to the order of integrability). These measures include the standard energy norm as a simple special case. A general approach is proposed to construct various measures based on using an auxiliary variational problem. Two classes of measures whose properties are close to those of the ${L}^{q}$ and ${L}^{\infty}$ norms are studied in more detail. Their properties are established, and explicitly computable two-sided estimates (minorants and majorants) that involve only known functions are constructed.

Key words: elliptic equations, estimates of the deviation from the exact solution, a posteriori estimates.

DOI: https://doi.org/10.31857/S0044466920050142


English version:
Computational Mathematics and Mathematical Physics, 2020, 60:5, 749–765

Bibliographic databases:

UDC: 519.63
Received: 28.10.2019
Revised: 28.10.2019
Accepted:14.01.2020

Citation: S. I. Repin, “Estimates of the deviation from exact solutions of boundary value problems in measures stronger than the energy norm”, Zh. Vychisl. Mat. Mat. Fiz., 60:5 (2020), 767–783; Comput. Math. Math. Phys., 60:5 (2020), 749–765

Citation in format AMSBIB
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\paper Estimates of the deviation from exact solutions of boundary value problems in measures stronger than the energy norm
\jour Zh. Vychisl. Mat. Mat. Fiz.
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\vol 60
\issue 5
\pages 767--783
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\crossref{https://doi.org/10.31857/S0044466920050142}
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\jour Comput. Math. Math. Phys.
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\pages 749--765
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