Simple algorithms for calculation of the classical a posteriori error estimates of numerical solutions of elliptic equations

V. G. Korneev

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 153 (2011) no. 4, pp. 11-27

Résumé

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.

Détail

Keywords: a posteriori estimates, error in approximate solutions, finite element method.

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