# Enhanced Upward Translations for Systems with Clusters
## 4 Numerical Results for Typical Kernels of BIE
### 2D Laplace BIE  
_Translation by means of weakly singular kernel \(W(x,y)\)._ In this case, the kernel \(K(x,y)\) in the conventional (4) and modified (9) translation is \(W(x,y)=-\frac{1}{2\pi k}\lvert\ln\lvert r\rvert\), where, \(k\) is the conductivity assumed unit (\(k=1\)); \(r=\lvert x-y\rvert\). Consider the quite unfavorable case, when the representative source is located at the relative distance \(\varepsilon=\left(R_{e}-\left|y_{clp}\right|\right)/R_{e}=0.2\) from the ES and the relative radius of the cluster is \(r_{cl}/R_{e}=0.12\). Figure 2 presents the comparative distributions of the absolute error along the angular coordinate (in radians) of the control surface. The maximal errorsoccur near the middle of an interval between two neighbouring control points. The minimal error is zero, as it must be when a control point coincides with a check point. For various distances \(\varepsilon\) from the ES and for various sizes \(r_{cl}\) of the cluster, the maximal and average errors are summarized in Table 1. It may be seen, that the accuracy of both conventional and modified translations decreases with decreasing distance \(\varepsilon\) and with growing radius \(r_{cl}\). Both Figure 2 and Table 1 evidently demonstrate a significant increase in the accuracy when using the modified translations as compared with the conventional approach. In particular, for sources located at the distance \(\varepsilon<0.25R_{\varepsilon}\), even for a cluster with a large relative radius \(r_{cl}/R_{\varepsilon}=0.12\), the accuracy of the modified translations is at least three-fold better than that of the conventional translations. _Translation by means of singular kernel \(G_{ny}(x,y)\)_. The kernel is \(G_{ny}(x,y)=-\frac{1}{2\pi}\frac{1}{r}r_{s}n_{ys}\) ([PERSON] summation convention is presumed). The kernel depends on the direction of the normal at a source point. The comparative distributions of errors along the control surface are provided in Figure 3. The maximal and average errors for various distances \(\varepsilon\) and radii \(r_{cl}\) of the cluster are summarized in Table 2. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & \multicolumn{3}{c}{**Conventional**} & \multicolumn{3}{c}{**Modified**} \\ \hline \(\varepsilon\) & **0.1** & **0.2** & **0.1** & **0.2** & **0.2** \\ \hline \(r_{cl}/R_{\varepsilon}\) & **0.07** & **0.07** & **0.12** & **0.07** & **0.07** & **0.12** \\ \hline
**max** & \(5.78\times 10^{-6}\) & \(3.13\times 10^{-6}\) & \(3.50\times 10^{-6}\) & \(8.32\times 10^{-7}\) & \(2.60\times 10^{-7}\) & \(9.06\times 10^{-7}\) \\ \hline
**average** & \(1.20\times 10^{-6}\) & \(6.37\times 10^{-7}\) & \(6.36\times 10^{-7}\) & \(1.43\times 10^{-7}\) & \(4.57\times 10^{-8}\) & \(1.59\times 10^{-7}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Maximal and average errors for translations by means of _weakly singular_ kernel (2D _singular_ Laplace BIE). Figure 2: Distribution of errors for conventional and modified translations by means of _weakly singular_ kernel (2D _singular_ Laplace BIE). Again, the results clearly show that the modified translations provide notable gains in accuracy. Translation by means of the hypersingular kernel \(H_{nx,ny}(x,y)\). The kernel is \(H_{nx,ny}(x,y)=-\frac{k}{2\pi}\frac{1}{r^{2}}\Big{(}n_{xj}n_{yj}-\frac{2}{r^{2}} r_{j}n_{xj}r_{s}n_{ys}\Big{)}\). The kernel depends on the direction of the normal at a source point. The comparative maximal and average errors are summarized in Table 3.  
Again, there is an order gain in the accuracy when using the modified translation. It reaches two orders for the distance \(\varepsilon=0.1\) and radius \(r_{cl}=0.07R_{e}\). Further decreasing the distance to \(\varepsilon=0.002\) and radius to \(r_{cl}/R_{e}=0.001\) provides additionally five orders gain in the accuracy of the modified method. Therefore, it may be expected that for a small radius \(r_{cl}\), the cluster of sources may be replaced with a single (representative) source. This option is discussed in the next section.