Simple algorithms for calculation of the classical a posteriori error estimates of numerical solutions of elliptic equations
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 153 (2011) no. 4, pp. 11-27
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As is well-known, for the problems in solid mechanics, “classical” approach to a posteriori error estimation stems from the Lagrange and Castigliano variational principles. If the problem is linear and an approximate solution satisfies geometrical restrictions, then potential energy of the error is estimated by the potential energy of the difference of the stress tensor corresponding to the approximate solution and any stress tensor satisfying the equations of equilibrium. We show that in many cases, construction of equilibrated stress fields can be done for a number of arithmetic operations, which is asymptotically optimal. This approach allows us also to improve known a posteriori estimates by means of arbitrary nonequilibrated tensors. Numerical experiments show that our a posteriori error estimators provide rather good efficiency indices, which often converge to unity, have linear complexity, and are robust.
Keywords:
a posteriori estimates, error in approximate solutions, finite element method.
@article{UZKU_2011_153_4_a1,
author = {V. G. Korneev},
title = {Simple algorithms for calculation of the classical a~posteriori error estimates of numerical solutions of elliptic equations},
journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
pages = {11--27},
publisher = {mathdoc},
volume = {153},
number = {4},
year = {2011},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/UZKU_2011_153_4_a1/}
}
TY - JOUR AU - V. G. Korneev TI - Simple algorithms for calculation of the classical a posteriori error estimates of numerical solutions of elliptic equations JO - Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki PY - 2011 SP - 11 EP - 27 VL - 153 IS - 4 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/UZKU_2011_153_4_a1/ LA - ru ID - UZKU_2011_153_4_a1 ER -
%0 Journal Article %A V. G. Korneev %T Simple algorithms for calculation of the classical a posteriori error estimates of numerical solutions of elliptic equations %J Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki %D 2011 %P 11-27 %V 153 %N 4 %I mathdoc %U http://geodesic.mathdoc.fr/item/UZKU_2011_153_4_a1/ %G ru %F UZKU_2011_153_4_a1
V. G. Korneev. Simple algorithms for calculation of the classical a posteriori error estimates of numerical solutions of elliptic equations. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 153 (2011) no. 4, pp. 11-27. http://geodesic.mathdoc.fr/item/UZKU_2011_153_4_a1/