# Detecting Inverse Boundaries by Weighted High-Order Gradient Collocation Method
## 4 Numerical Implementation
### Multiply Connected Domains: Type One  
Referring to Figure 10(a)-d, four multiply connected domains equipped with various boundary conditions are in consideration: half unknown inner boundary, two half unknown inner boundaries, unknown inner boundary, and half unknown outer boundary. The domain is enclosed by a circular perimeter with radius \(R_{o}=1\); each inner hole has a radius equal to 0.2. Again, the domain is governed by the Laplace equation while a third-order approximation is adopted. The general expressions for the boundary conditions on the outer boundary in the four configurations are described by  
\[\begin{split}& u(R_{o},\theta)=\left[-R_{o}^{3}\sin(3\theta)+R_{o}^{2 }\cos(2\theta)\right]\left(1+\frac{s}{100}\times rand\right),\,\beta_{1}\leq \theta\leq\beta_{2}\\ & u_{n}(R_{o},\theta)=\left[-3R_{o}^{2}\sin(3\theta)+2R_{o}\cos(2 \theta)\right]\left(1+\frac{s}{100}\times rand\right),\,\beta_{1}\leq\theta \leq\beta_{2}\end{split} \tag{37}\]  
where \(\beta_{i}\) has the same definition as stated previously. For \(s=0\), \(u\) on the missing boundary should satisfy the analytical solution in the inverse analysis, which is  
\[u=-R^{3}\sin(3\theta)+R^{2}\cos(2\theta) \tag{38}\]  
As depicted in Figure 10(a)-d, the same set of source points and collocation points are adopted in the discretization of four configurations, i.e., \(N_{s}=N_{c}=470\). For Figure 10(a)-c, RK support size is \(a=3.3d\). For Figure 10(d), RK support size is \(a=3.5d\); further, it is noted that the points on the circumference have been spun with a small angle to avoid overlaps of points, as can be observed in the configuration. The contour plots of absolute errors (%) are shown in Figure 11(a)-d; it is observed that large errors occur along the missing boundary, and the distribution of errors exhibits symmetry as can be expected due to symmetric configurations of the problems. Except the case shown in Figure 11(d), the absolute errors are quite small. The use of quadratic basis in the formulation of the RK and HG-RK shape functions is able to yield the desired accuracy in the approximation of the cubic polynomial. Concerning the percentage of disturbance \(s\) added on the outer boundary conditions, Figure 12(a)-d compares the numerical solutions with analytical ones, and good agreement is observed even when \(s=3\), except the difficult case of Figure 10(d).  
Figure 10: Comparison of CPU time by different methods in the annular domain.  
Figure 11: Discretization of type one multiply connected domains: (**a**) half unknown inner boundary; (**b**) two half unknown inner boundaries; (**c**) unknown inner boundaries; (**d**) half unknown outer boundary.  
Figure 12: _Cont._  
Table 2 summarizes the maximal absolute errors (%) for \(u\), \(u_{x}\), and \(u_{y}\) obtained by various \(s\) added on the outer boundary conditions. Again, high accuracy in the approximation is reached, excluding the difficult case of Figure 11d. As such, the stability of the method is demonstrated. Referring to Figure 14, it is observed explicitly that HG-RKCM is the most efficient method.  
Figure 12: Contour plots of absolute errors (%): (**a**) half unknown inner boundary; (**b**) two half unknown inner boundaries; (**c**) unknown inner boundaries; (**d**) half unknown outer boundary.  
Figure 13: Comparison of numerical solutions along missing boundary obtained by various \(s\): (**a**) half unknown inner boundary; (**b**) two half unknown inner boundaries; (**c**) unknown inner boundaries; (**d**) half unknown outer boundary.