  The present paper develops a variational theory of discrete fields defined on
abstract cellular complexes. The discrete formulation is derived solely from a
variational principle associated to a discrete Lagrangian density on a discrete
bundle, and developed up to the notion of symmetries and conservation laws for
solutions of the discrete field equations. The notion of variational integrator
for a Cauchy problem associated to this variational principle is also studied.
The theory is then connected with the classical (smooth) formulation of
variational field theories, describing a functorial method to derive a discrete
Lagrangian density from a smooth Lagrangian density on a Riemannian fibered
manifold, so that all symmetries of the Lagrangian turn into symmetries of the
corresponding discrete Lagrangian. Elements of the discrete and smooth theories
are compared and all sources of error between them are identified. Finally the
whole theory is illustrated with the discretization of the classical
variational formulation of the kinematics of a Cosserat rod.
