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 irregu lar lattice, using discrete field quantities roughly analogous to different ial forms. The role of the space Lambda(p) of p-forms at a point is taken o n by the linear space generated at a graph vertex by the geometrical p-simp lices which contain it. The vector operations divergence, gradient, and cur l are developed using the boundary delta and coboundary d. Dot, cross, and scalar products are defined in such a way that discrete analogs of the vect or integral theorems, including theorems of Gauss-Ostrogradski, Stokes, and Green, as well as most standard vector identities hold exactly, not as app roximations to a continuum limit. Physical conservation laws for the models become theorems satisfied by the discrete fields themselves. Three discret e 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 latt ice. Green functions are developed using a generalized Helmholtz decomposit ion of the fields. [S1063-651X(99)09801-3].