# Enhanced Upward Translations for Systems with Clusters
## 6 Summary  
The paper is concerned with using BEM for accurate evaluation of local fields arising near concentrators in structures composed of multiple elements interacting on their contacts. It presents theoretical rationale and a method to increase the accuracy of matrix-to-vector multiplications, performed on iterations of KI-FMM when solving the BEM system. The method improves the accuracy of fields translated from those sources of the BEM system, whose far fields, when translated conventionally, have the greatest errors. These leaf sources are assembled into clusters by choosing an appropriate leading source, and by grouping around it the sources, located closer than a given threshold. The latter is found in numerical experiments performed for typical kernels. A composed cluster is characterized by the representative source. Its intensity, location and normal (for kernels depending on the normal at the source point) are uniquely defined by the leaf sources comprising it. For a cluster distinguished, the accuracy of the conventional approximation of its far field is increased by the modified S2M translation suggested. The improved translation is based on the approach \"subtract and add\". The approach consists of subtracting and adding the directly calculated input from the representative source, which generates the dominant part of the far field. We define the residual far field as the difference between the true far field from the cluster sources and the far field of the representative source. It decreases much faster than the far field from the sources. This yields that the maximal error of its translation is notably less than that of the conventional approximation of the far field from the sources themselves. The approximation of the residual far field is performed by employing the modified kernel adjusted to its rapid decrease with growing distance. The modified equivalent intensities are found from the algebraic system obtained by equating the approximated residual field to its exact values at check nodes. As usual for the \"subtractand add\" approach, the gain is due to the exact evaluation of the added principal term. Numerical results for typical kernels of BIE solved by BEM demonstrate the notable increase in accuracy when employing the modified S2M translations. Composing the clusters, building and inversion of the modified matrices are performed only once before starting iterations. The complexity of modified translations, performed on iterations, is the same as that of conventional translations. As a result, the generation of clusters and using modified translations for improving the accuracy of S2M translations does not increase the time expense. Furthermore, to the accuracy acceptable for many practical applications, the residual far field may be neglected as compared with the dominant far field of the representative source. This excludes the need to solve the modified system and, consequently, in the evaluation and inversion of the matrices for each of the clusters. Thus, the modified translations are actually reduced to calculation far fields from the representative sources at the nodes on the parent check surface. This additionally notably simplifies the modified S2M translations. The same simplification is reasonable to use when the number \(N_{L}\) of sources in a leaf is less, close or insignificantly greater than the number \(N_{e}\) of nodes on the ES of the leaf. In these practically important cases, each of the sources may be considered as a cluster, comprised of a single source. Then there is no sense in finding \(N_{e}\) equivalent intensities. The needed values of the total far field are _exactly_ calculated as the sums of fields arising from the \(N_{L}\) sources. Then the time expense is less (when \(N_{L}<N_{e}\)) or on the level (when \(N_{L}\sim N_{e}\)) of that for calculations involving equivalent intensities. The method suggested is also applicable to improving the accuracy of M2M upward translations. They translate far fields from children to their parents.