# Enhanced Upward Translations for Systems with Clusters
## 4 Numerical Results for Typical Kernels of BIE  
### Computational Prerequisites  
Consider kernels entering the singular and hypersingular BIE of the Laplace and Navier equations. To keep track of the results of the paper [24], where the accuracy was studied for conventional translations, we shall use the notations and parameters of the cited paper. Thus, in this section, when specifying a kernel, we use normal (not bold) typescript for matrices and vectors. The starting singular and hypersingular BIE are  
\[\int_{S}W(x,y)\Delta q_{ny}(y)dS_{y}+\int_{S}G_{ny}(x,y)\Delta u(y)dS_{y}=0.5u( x),\ x\in S \tag{10}\]  
\[\int_{S}Q_{nx}(x,y)\Delta q_{ny}(y)dS_{y}+\int_{S}H_{nx,ny}(x,y)\Delta u(y)dS_{ y}=0.5q_{nx}(x),\ x\in S \tag{11}\]  
Herein, \(S\) is a surface (open or closed), on which the potential or/and the flux may be discontinuous. The surface is assumed smooth enough to have the classical factor 0.5 on the right-hand side. \(W(x,y)\) is a _weakly singular_ kernel. \(G_{ny}(x,y)\) is a field of double-layer potential; the subscript \(ny\) indicates that the kernel depends on the normal to the surface \(S\) at an integration point \(y\); the dependence on the normal is of prime significance for the further analysis of the accuracy. \(Q_{nx}(x,y)\) is the singular kernel; similar to \(W(x,y)\), it does not depend on the normal at the integration point. \(H_{nx,ny}(x,y)\) is the _hypersingular_ kernel; similar to \(G_{ny}(x,y)\), it depends on this normal; \(q_{ny}\) is the normal component of the flux \(q\) on \(S\). \(\Delta q_{ny}=q_{ny}^{+}-q_{ny}^{-}\) is the discontinuity of the flux at the integration point \(y\); \(\Delta u=u^{+}-u^{-}\) is the discontinuity of the potential \(u\). The sign \"plus\" (\"minus\") marks the limiting values from the side with respect to which the normal is outward (inward). For external boundaries, the normal is assumed outward. Then, for a given potential \(u\) (flux component \(q_{ny}\)) on it, we set \(u^{+}=u\), \(u^{-}=0\) (\(q_{ny}^{+}=q_{ny}\), \(q_{ny}^{-}=0\)). For the Laplace equation, the potential \(u\) is scalar; for the Navier equation, it is a vector. Similar to [24], in the calculations discussed below, in \(2D\), the leaf cell is a square with half-sides of unit length. The equivalent surface (ES) is a circle of radius \(R_{e}=1.415\). The radii of the check (ChS) and control surfaces are the same \(R_{Ch}=R_{Cont}=2.98\). The nodes at all surfaces are located uniformly; their numbers are \(N_{nd}=24\) for the ES and ChS; and \(N_{Cont}=1000\) for the control surface. In 3\(D\), the leaf cell is a cube with sides of unit length; the ES is spherical with a radius \(R_{e}=0.866\). For the _Laplace_ equation, the radii of the check (ChS) and control surfaces are the same \(R_{Ch}=R_{Cont}=1.49\). The nodes on them are located almost uniformly; their numbers are \(N_{nd}=218\) for the ES and ChS. The nodes at the control surface are taken in the cross-section coinciding with the plane of axes \(x_{1}\) and \(x_{2}\); their number in this cross-section is \(N_{Cont}=1580\). For the Laplace equation, the translated field is a scalar, which becomes zero at some points of the control surface. Thus the maximal _absolute_ errors are studied. For the _Navier_ equation, the number of translated components is three-fold greater than for the 3D Laplace BIE. To maintain the accuracy, larger number of nodes \(N_{nd}=386\) is used for the ES and ChS. The radius of the ChS is \(R_{Ch}=1.49\). That of the control surface is \(R_{Cont}=2.00\). The control points are uniformly located at each of the 18 principle circles. The normal to a circle plane is taken orthogonal to the \(x_{2}\)-axis, its angle with the \(x_{1}\)-axis is changed with the step \(10^{\circ}\). Along a principle circle, there are 72 control points. Thus the total number of control points is \(18\cdot 72=1296\). For the Navier equation, the translated quantities are vectors which in the examples considered are non-zero.