Vector difference calculus for physical lattice models

A vector difference calculus is developed for physical models defined on a general triangulating graph G, which may be a regular or an extremely irregular lattice, using discrete field quantities roughly analogous to differential forms. The role of the space Λᵖ of p-forms at a point is taken on by the linear space generated at a graph vertex by the geometrical p-simplices which contain it. The vector operations divergence, gradient, and curl are developed using the boundary {∂} and coboundary $d.$ Dot, cross, and scalar products are defined in such a way that discrete analogs of the vector integral theorems, including theorems of Gauss-Ostrogradski, Stokes, and Green, as well as most standard vector identities hold exactly, not as approximations to a continuum limit. Physical conservation laws for the models become theorems satisfied by the discrete fields themselves. Three discrete lattice models are constructed as examples, namely a discrete version of the Maxwell equations, the Navier-Stokes equation for incompressible flow, and the Navier linearized model for a homogeneous, isotropic elastic medium. Weight factors needed for obtaining quantitative agreement with continuum calculations are derived for the special case of a regular triangular lattice. Green functions are developed using a generalized Helmholtz decomposition of the fields.