Erratum to: A Stabilized Mixed Finite Element Method for Elliptic Optimal Control Problems

Erratum to: J Sci Comput DOI 10.1007/s10915-015-0050-3

The authors would like to correct the numerical results of Section 6 published in the original version as follows:

Example 6.1

For the first example, we take the stabilized parameters $\mu =0.2$ and $\delta =0.5$.

Piecewise linear $C^0$ element is adopted for the approximation of the flux $\sigma$. The numerical results and main CPU time are presented in Table 1, and numerical plots for the control, the state, and the flux state are shown in Figs. 1, 2, and 3, respectively.

Table 1 Example 1 with piecewise linear $C^0$ elements for $\sigma _h$

Fig. 1

Approximate control $u_h$ (left) and its contour line (right) for Example 1 with $\mu =0.2$ and $\delta =0.5$

Fig. 2

Approximate state $y_h$ for Example 1 with $\mu =0.2$ and $\delta =0.5$

Example 6.2

For the second example, we consider the stabilized parameters $\mu =\delta =0.5$.

Table 2 shows that a first-order convergence is obtained for the control, which is well matched with the theoretical analysis. Figures 4, 5, and 6 show the approximate profiles of the control, the state, and the flux state, respectively, when the lowest order RT element is adopted for the approximation of the flux $\sigma$.

Fig. 3

Approximate flux state $\sigma _h$ for Example 1 with $\mu =0.2$ and $\delta =0.5$

Table 2 Example 2 with the lowest order RT elements for $\sigma _h$

Fig. 4

Approximate control $u_h$ (left) and its contour line (right) for Example 2 with $\mu =\delta =0.5$

Fig. 5

Approximate state $y_h$ for Example 2 with $\mu =\delta =0.5$

Fig. 6

Approximate flux state $\sigma _h$ for Example 2 with $\mu =\delta =0.5$

Both numerical examples support the theoretical analysis very well.