# Detecting Inverse Boundaries by Weighted High-Order Gradient Collocation Method
## 4 Numerical Implementation
### Multiply Connected Domains: Type Two  
As shown in Figure 15a-d, four multiply connected domains having various boundary conditions are considered: unknown upper inner boundary, three half unknown inner boundaries, unknown inner boundaries, and half unknown outer boundary. The outer radius \(R_{o}\) of the domain is 1.0 while the inner radius in each hole is 0.2. The governing equation and boundary conditions are the same as the ones described in Section 4.3. For the cases shown in Figure 15a-c, \(N_{s}=N_{c}=467\) and RK support size is \(a=3d\). For the case shown in Figure 15d, a half layer of ghost points is arranged outside the upper boundary with a distance \((1+1.25d)\) from the point \((0,0)\) in the domain in order to reach desired accuracy; thus, \(N_{s}=N_{c}=498\) and RK support size is \(a=3.9d\). The contour plots of absolute errors (%) are shown in Figure 16a-d, in which the locations of large errors are consistent with the locations of unknown boundaries, and symmetric distribution of errors is explicitly observed. The overall trend of absolute errors is small. Then, the disturbance \(s=0,1,2,3\) is added to the outer boundary conditions as given in Equation (37). Figure 17a-d compares the numerical solutions with analytical solutions, and good agreement is observed even when \(s=3\), except the case shown in Figure 17d. The maximal absolute errors (%) for \(u\), \(u_{x}\), and \(u_{y}\) obtained by various \(s\) on the unknown boundaries are summarized in Table 3. It is observed that Figure 15d is the most difficult case, where  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(s\) & **0** & **1** & **2** & **3** \\ \hline \multirow{3}{*}{(**a**)} & \(u\) & 0.3942 & 0.5251 & 0.5947 & 0.8998 \\  & \(u_{x}\) & 4.1349 & 4.5853 & 4.9640 & 5.2201 \\  & \(u_{y}\) & 3.0719 & 3.3959 & 3.9454 & 4.1871 \\ \hline \multirow{3}{*}{(**b**)} & \(u\) & 0.3899 & 0.7809 & 0.9303 & 1.1415 \\  & \(u_{x}\) & 3.4197 & 5.2031 & 5.5692 & 8.8528 \\  & \(u_{y}\) & 3.1640 & 3.7078 & 4.0539 & 8.7244 \\ \hline \multirow{3}{*}{(**c**)} & \(u\) & 0.3932 & 0.6791 & 1.0372 & 1.6399 \\  & \(u_{x}\) & 4.0919 & 4.6996 & 5.0791 & 6.5832 \\  & \(u_{y}\) & 3.0315 & 3.3222 & 3.9394 & 4.4157 \\ \hline \multirow{3}{*}{(**d**)} & \(u\) & 4.6230 & 9.7452 & 11.6293 & - \\  & \(u_{x}\) & 29.5079 & 32.3523 & 39.8299 & - \\ \cline{1-1}  & \(u_{y}\) & 27.6338 & 35.6646 & 41.2359 & - \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of maximal absolute errors (%) obtained by various \(s\): (a) half unknown inner boundary; (b) two half unknown inner boundaries; (c) unknown inner boundaries; (d) half unknown outer boundary.  
Figure 14: Comparison of CPU time by different methods in the multiply connected domains: type one.  
\(u_{x}\) and \(u_{y}\) are sensitive to the disturbance. Apart from this, the stability of the method is demonstrated. As shown in Figure 18, HG-RKCM is the most efficient method among the three methods.  
Figure 16: _Cont._  
Figure 15: Discretization of type two multiply connected domains: (**a**) unknown upper inner boundary; (**b**) three half unknown inner boundaries; (**c**) unknown inner boundaries; (**d**) half unknown outer boundary.  
Figure 16: Contour plots of absolute errors (%): (**a**) unknown upper inner boundary; (**b**) three half unknown inner boundaries; (**c**) unknown inner boundaries; (**d**) half unknown outer boundary.  
Figure 17: Comparison of numerical solutions along missing boundary obtained by various \(s\): (**a**) unknown upper inner boundary; (**b**) three half unknown inner boundaries; (**c**) unknown inner boundaries; (**d**) half unknown outer boundary.