# 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:

- 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.
- 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.