The Cell Method as a Case of Bialgebra

The Cell Method (CM) associates any physical variable with the geometrical and topological features, usually neglected by the differential formulation. This goal is achieved by abandoning the habit to discretize the differential equations. The governing equations are then derived in algebraic manner directly, by means of the global variables. In the original formulation of the CM, the association between physical variables and geometry is made on the basis of physical considerations. In this paper, we analyze the same association under the mathematical point of view. This allows us to view the CM as a geometric algebra, which is an enrichment of the exterior algebra. The elements and their inner and outer orientations are derived inductively. They are obtained from the outer product of the geometric algebra and the features of . Space and time global variables are treated in a unified four-dimensional space/time cell complex, whose elementary cell is the tesseract. Moreover, the configuration variables with their topological equations, on the one hand, and the source variables with their topological equations, on the other hand, are viewed as a bialgebra and its dual algebra.