# Enhanced Upward Translations for Systems with Clusters
## 1 Introduction  
A wide range of practical problems, concerning strength and safety, require accurate evaluation of local fields. Such are problems of rock and fracture mechanics, for which extreme, rather than average, values of physical fields are of prime significance (e.g., [1; 2; 3; 4; 5; 6]). These problems are characterized by a strongly inhomogeneous distribution of flaws, in particular, faults, cracks, inclusions of various sizes and mutual location. Individual flaws of large sizes, and collections of closely located flaws, are normally the sources of the most dangerous events. Such flaws may be related to clusters. They strongly influence not only the safety of engineering constructions but also the accuracy of their numerical modeling. The present paper tends to suggest a means to maintain accuracy. In view of the mentioned applications, we shall have in mind, first of all, _static_ 2D and 3D problems involving Laplace and Navier equations. When accounting for local fields in inhomogeneous structures with elements interacting on multiple contacts, the boundary integral equations (BIE) appear to be an appropriate means (e.g., [7; 8; 9; 10; 11; 12]). Their special forms reduce a mathematical problem to boundaries of structural elements and contain as densities the very quantities which describe contact conditions. The BIEs are solved by the boundary element methods (BEM), which employ discretization of boundaries onto a set of boundary elements with approximation densities at each of them (e.g., [8; 9; 10; 11; 12]). The main drawback of a BEM is that the matrix of its interaction coefficients is fully populated. Consequently, each of the repeatedly used operations of matrix-to-vector multiplications has the complexity \(N^{2}\). With growing \(N\), the time expense becomes enormous. Nowadays, this drawback is smoothed by employing speciallydesigned methods. They decrease the complexity of matrix-to-vector multiplication to the level usual for finite element methods (e.g., [13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25]). The first such method was suggested by [PERSON] [13]. It employs _a priori_ geometrical information on element locations, and the analytical expression for an integral _kernel_. Details of the method called the analytical fast multipole method (A-FMM), may be found in the textbook [19]. Of essence, it employs the geometrical data to build the hierarchical tree. The analytical expression for a kernel is used for its multipole expansion and for organizing translations of fields in up-downward runs over the tree. Thus, the method is _kernel dependent_. It performs the matrix-to-vector multiplication with the computing complexity to at most \(O(N\ logN)\). The other breakthrough method is the adaptive cross approximation (ACA) algorithm by [PERSON] [16]. In contrast with the A-FMM, the ACA algorithm is _purely algebraic_. It is _independent_ of the underlying formulation, in particular, on the knowledge of integral kernel and geometrical features of a physical system. The latter are accounted for implicitly by appropriate algebraic manipulations. In applications to static and low-frequency problems involving asymptotically smooth (non-oscillatory) kernels, the ACA algorithm has shown the complexity \(O(N\ logN)\). Later on, it was successfully applied to problems of electromagnetic scatter and radiation (e.g., [26; 27]) for which the kernels are less smooth. Clearly, however, the geometrical information is of great value for speeding calculations in static and low-frequency problems of fluid and solid mechanics. The mentioned problems of rock and fracture mechanics are among them. It is reasonable to make use of _a priori_ geometrical information in a way that avoids involved analytical work for each particular kernel. The method suggested in the paper [18] was performed in a kernel-independent form. The method follows the same line as the A-FMM in building the tree and up-downward runs over it. The difference is in the design of translations.