OOF2: The Manual

Name

Z-Z Estimator (ZZ_Estimator) — Error estimation by Zienkiewicz and Zhu's superconvergent patch recovery.

Synopsis

ZZ_Estimator(norm,flux,threshold)

Details

  • Base class: ErrorEstimator
  • Parameters:

    norm
    How to measure the size of an error. Type: An object of the ErrorNorm class.
    flux
    Type: A Flux object.
    threshold
    Maximum allowable percentage error. Type: A real number in the range [0, 100].

Description

ZZ_Estimator is an ErrorEstimator used when modifying a Mesh by adaptive mesh refinement. It uses the Superconvergent Patch Recovery technique of Zienkiewicz and Zhu.[38]

The error of the finite element approximation \(u_h\) with respect to the exact solution \(u\) is


  \[e = u - u_h\]
      (6.165)

and the error of the flux (proportional to gradients of \(u\)) is


\[e_\sigma = \sigma - \sigma_h\]
      (6.166)

These definitions are nearly useless, however, because the exact solutions \(u\) and \(\sigma\) 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 \(\sigma_h\) computed from the finite element solutions \(u_h\) 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 \(\sigma^*\) of the stress \(\sigma\) at a node by interpolating between the stresses at the superconvergent points in a small set of elements (a patch) around the node.

The ZZ_Estimator compares the recovered and directly computed values of the given flux and refines those in which the norm of the relative error \((\sigma-\sigma^*)/\sigma^*\) is greater than the given threshold (given as a percentage).

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

Element Type Linear Quadratic
Triangle
Quadrilateral

Figure 6.117 and Figure 6.118 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.

Figure 6.117. Normal Patch Recovery

Normal Patch Recovery

A Typical patch for the Superconvergent Patch Recovery procedure.


Figure 6.118. Boundary Patch Recovery

Boundary Patch Recovery

A Boundary patch for the Superconvergent Patch Recovery procedure.




[38]

  1. 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)

  2. 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)