Algorithm 1

1) Choose $u_h^0\in U_{ad}$ and set $k=0$ ;

2) Compute $y_h^k$ by solving $a_h\left(y_h^k\mathrm{,}{I}_h^{\ast }v\right)=\left(f\mathrm{,}{I}_h^{\ast }v\right)+{\left(B{u}_h^k\mathrm{,}{I}_h^{\ast }v\right)}_{\Gamma }$ ;

3) Compute $z_h^k$ by solving $a\left(z_h^k\mathrm{,}{I}_h^{\ast }{w}_h\right)=\left(y_h^k-y_{\Omega }\mathrm{,}{I}_h^{\ast }w\right)$ ;

4) Set $u_h^{k+1}=P_{\left[u_a\left(x\right),u_b\left(x\right)\right]}\left(-\frac{B^{\ast }{z}_h^k}{\lambda }\right)$ ;

5) If $\left|u_h^{k+1}-u_h^k\right|\le {10}^{-6}$ , then we set ${\overline{u}}_h=u_h^{k+1}$ , else $k=k+1$ , and go back to Step 2.