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Two-sided a posteriori error estimates for mixed formulations of elliptic problems


Repin, S; Sauter, S; Smolianski, A (2007). Two-sided a posteriori error estimates for mixed formulations of elliptic problems. SIAM Journal on Numerical Analysis, 45(3):928-945.

Abstract

The present work is devoted to the a posteriori error estimation for mixed approximations of linear self-adjoint elliptic problems. New guaranteed upper and lower bounds for the error measured in the natural product norm are derived, and individual sharp upper bounds are obtained for approximation errors in each of the physical variables. All estimates are reliable and valid for any approximate solution from the class of admissible functions. The estimates contain only global constants depending solely on the domain geometry and the given operators. Moreover, it is shown that, after an appropriate scaling of the coordinates and the equation, the ratio of the upper and lower bounds for the error in the product norm never exceeds 3. The possible methods of finding the approximate mixed solution in the class of admissible functions are discussed. The estimates are computationally very cheap and can also be used for the indication of the local error distribution. As applications, the diffusion problem as well as the problem of linear elasticity are considered.

Abstract

The present work is devoted to the a posteriori error estimation for mixed approximations of linear self-adjoint elliptic problems. New guaranteed upper and lower bounds for the error measured in the natural product norm are derived, and individual sharp upper bounds are obtained for approximation errors in each of the physical variables. All estimates are reliable and valid for any approximate solution from the class of admissible functions. The estimates contain only global constants depending solely on the domain geometry and the given operators. Moreover, it is shown that, after an appropriate scaling of the coordinates and the equation, the ratio of the upper and lower bounds for the error in the product norm never exceeds 3. The possible methods of finding the approximate mixed solution in the class of admissible functions are discussed. The estimates are computationally very cheap and can also be used for the indication of the local error distribution. As applications, the diffusion problem as well as the problem of linear elasticity are considered.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:a posteriori estimate, two-sided bounds, mixed approximation, elliptic problem
Language:English
Date:2007
Deposited On:02 Nov 2009 12:56
Last Modified:21 Nov 2017 14:21
Publisher:Society for Industrial and Applied Mathematics
ISSN:0036-1429
Additional Information:Copyright © 2007, Society for Industrial and Applied Mathematics
Publisher DOI:https://doi.org/10.1137/050641533
Related URLs:http://www.math.uzh.ch/fileadmin/math/preprints/21-05.pdf

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