OOF2: The Manual
Z-Z Estimator (ZZ_Estimator) — Error estimation by Zienkiewicz and Zhu's superconvergent patch recovery.
The error of the finite element approximation with respect to the exact solution is
and the error of the flux (proportional to gradients of ) is
These definitions are nearly useless, however, because the exact solutions and are almost always unknown. It is therefore necessary to invent a way of estimating the error.
The error estimate procedure developed by Zienkiewicz and Zhu, employing Superconvergent Patch Recovery has proved to be effective and economical both in evaluating errors and driving adaptive mesh refinement.
It is well known that the field derivatives and fluxes computed from the finite element solutions do not possess interelement continuity and have a low accuracy at nodes and element boundaries. However, there are points within elements, known as superconvergent points, at which the derivatives and fluxes are more accurate than at any other point within the element: values at the superconvergent points converge more quickly to the true values as the element size decreases. It is possible to get (or recover) a more accurate value of the stress at a node by interpolating between the stresses at the superconvergent points in a small set of elements (a patch) around the node.
ZZ_Estimator compares the recovered and
directly computed values of the given
refines those in which the
norm of the relative
is greater than the given
threshold (given as a
The remainder of this page is not required reading.
The superconvergent points for various element types are summarized in Table 6.7, where the black dots represent superconvergent points.
Table 6.7. Superconvergent Points of Various Elements Types
Figure 6.116 and Figure 6.117 illustrate typical patches. Only the values of nodes that are completely inside the patch are recovered, unless the patch contains boundaries, in which case values at the edge of the patch are recovered too.
O.C. Zienkiewicz and J.Z. Zhu, The Superconvergent patch recovery and a posteriori error estimators. Part 1. The recovery technique, Int. J. Numer. Methods Eng., 33, 1331-1364 (1992)
O.C. Zienkiewicz and J.Z. Zhu, The Superconvergent patch recovery and a posteriori error estimators. Part 2. Error estimates and adaptivity, Int. J. Numer. Methods Eng., 33, 1365-1382 (1992)