A posteriori error estimation

Now, we will consider a posteriori error estimates for the discretization process described above, i.e., for the error in the displacement,  u - u_h , and for the error in the corresponding stresses,  $\sigma$ - $\sigma_{h}^{}$ . We begin with two of the traditional strategies for error estimation.

1) The ZZ-approach: The error indicator proposed by Zienkiewicz and Zhu [22] for finite element models in structural mechanics is based on the idea of higher-order stress recovery by local averaging. The element-wise error  |$\sigma$ - $\sigma_{h}^{}$|_K  is thought to be well represented by the auxiliar quantity  $\eta_{K}^{}$ : = |$\cal {M}$_h$\sigma_{h}^{}$ - $\sigma_{h}^{}$|_K , where  $\cal {M}$_h$\sigma_{h}^{}$  is a local (super-convergent) approximation of  $\sigma$ . The corresponding (heuristic) global error estimator reads

$$\sigma \sigma_{h}^{} \approx \eta_{ZZ}^{} \Big( \sum_{K\in\mathbb T_h}^{} \cal {M} \sigma_{h}^{} \sigma_{h}^{} \Big)^{1/2}_{}$$ (47)

For our purpose we assume the discrete stresses to be constant over each cell. One possible construction of $\cal {M}$_h$\sigma_{h}^{}$ is the patch-wise L²-projection P_K$\sigma_{h}^{}$ onto the space of (bi-)linear shape functions. Here the nodal value at a point of the triangulation determining $\cal {M}$_h$\sigma_{h}^{}$ is obtained by averaging the cell-wise constant values of $\sigma_{h}^{}$ of those cells having this point in common. For cells containing hanging nodes this process is appopriately modified (see Figure 2).

Abbildung 2: Sketch: Averaging process for ZZ-Indicator

2) An energy error estimator: Johnson and Hansbo [11] proposed an error estimator for the primal-mixed formulation of the Hencky model which is based on monotinicity properties of the energy form and, under some additional heuristic assumptions, bounds the error in the global energy norm. Let  $\Omega^{e}_{h}$  and  $\Omega^{p}_{h}$  denote the union of elements where the discrete solution behaves elastic and plastic, respectively. Then, the estimator reads

$$\sigma \sigma_{h}^{} \approx \eta_{E}^{} \Big( \sum_{K\in\mathbb T_h}^{} \eta_{K}^{2} \Big)^{1/2}_{}$$ (48)

with the local error indicators

$$\eta_{K}^{2} \begin{cases} \,h_k^4 \max_K \{\vert R(u_h)\vert\}^2 \,, &\t... ...h)\vert\,dx \,, &\text{if} \;\; K\in \Omega^p_h \,, \end{cases}$$

where on each element  K $\in$ $\mathbb {T}$_h the local residual is defined by

R(u_h)  : =  | divC$$\epsilon$$(u_h)| + $${\textstyle\tfrac{1}{2}}$$h_K^-1|[n ^. C$$\epsilon$$(u_h)]| .

Here, C_i is some interpolation constant usually set to one. This estimator is rather heuristic, as it relies on the assumption that the plastification zone is already correctly captured on the current mesh. Furthermore, it is of only sub-optimal order in the plastic zone which results in mesh over-refinement in $\Omega^{p}_{}$, though the stresses are suspected to be rather smooth there. The ZZ-estimator (39) does not suffer from this deficiency as it essentially relies on the smoothness of  $\sigma$. Hence, we are led to modify the estimator (40) by replacing the obviously too crude a bound  max_K| C$\epsilon$(u_h)|  in the plastic zone by  max_K| C$\epsilon$(u_h) - $\cal {M}$_hC$\epsilon$(u_h)| . This gives us the local (still heuristic) error indicators

$$\eta_{K}^{2} \begin{cases} \,h_k^4 \max_K \{\vert R(u_h)\vert\}^2 \,, &\t... ...h)\vert\,dx \,, &\text{if} \;\; K\in \Omega^p_h \,. \end{cases}$$

3) The weighted local error estimator: Finally, we recall our weighted a posteriori error estimator for the primal formulation (27) of the Hencky problem,

$$\approx \sum_{K\in\mathbb T_h}^{} \tilde{\omega}_{K}^{} \rho_{K}^{}$$ (49)

with the local residuals

$$\rho_{K}^{} \nabla \nabla \partial$$

and the weights

$$\tilde{\omega}_{K}^{} \nabla^{2}_{h} \tilde{z}_{h}^{}$$

obtained from the approximate dual solution  $\tilde{z}_{h}^{}$ .