Simple and efficient a posteriori error estimators

Vadim Korneev

June 3, 2008, 1:30 p.m. T 1010

Numerical experiments show that a posteriory estimators based on the use of fluxes (stresses) obtained by projection of FE fluxes into the set of the balanced (equilibrated) ones can be quite efficient. In this lecture, we consider two ways of numerical evaluation of such fluxes:

  1. Direct evaluation, e.g., by interpolation of the average nodal FE fluxes and then correcting interpolated averaged fluxes by means of balance (equilibrium) equations. Last operation requires computation of some 1-D integrals.

  2. Evaluation of the exactly balanced fluxes by solving the dual problem. The key step is the choice of self-balanced coordinate vectors of fluxes, which in some cases makes the system of algebraic equations similar to the FE one for the primal problem.

The approach allows us also to derive general guaranteed a posteriory error estimate for second order elliptic problems, which do not contain any constants and are easily computable for a wide range of problems.

We present results of numerical experiments. In all our experiments with the 1-st type algorithms they produced convergent effectiveness indices. The 2-nd type algorithms provided rather good effectiveness indices, which, however, did not converge.