Some new findings on the mathematical structure of the cell method

In the classification diagram of the Cell Method (CM), which is the truly algebraic numerical method, the global variables are stored in two columns: the column of the configuration variables, with their topological equations, and the column of the source variables, with their topological equations. The structure of the classification diagram is the same for both the global and the field variables of every physical theory of the macrocosm. The importance of this diagram stands just in its ability of providing a concise description of physical variables, without distinguishing between the physical theories. Recently, we have shown that we can provide the classification diagram of the CM with a mathematical meaning, in addition to a physical meaning. Actually, we can recognize in the classification diagram of the CM a structure of bialgebra. In this paper, we give a further insight into the mathematical foundations of the CM by comparing the structure of the algebraic formulation with the structure of the differential formulation. Particular attention is devoted to the computation of limits, by highlighting how the numerical techniques used for performing limits may imply a loss of information on the length scales associated with the solution. Since the algebraic formulation does not make use of the limit process, this means that the algebraic formulation preserves the information on the length scales associated with the solution. Conversely, the differential formulation is forced to introduce a proper enrichment of the equations and/or the space of reals for taking into account the length scales associated with the solution. © 2015, North Atlantic University Union NAUN. All rights reserved.