First variation formula for discrete variational problems in two independent variables

We introduce a geometric setup for discrete variational problems in two independent variables based on the theory of bundles modelled on cell complexes, and characterize geometrically a first variation formula and a discrete Noether theorem for symmetries of the discrete Lagrangian. We explore the existence of discrete variational integrators, which will conserve the discrete Noether currents in the theory. The range of applications of the theory includes discrete versions of variational problems with or without PDE constraints and energy-conserving integrators in mechanics of continua. We illustrate the theory with a variational integrator for the movement of a string.