  Magnetic flux penetration in superconductors involves a rich variety of
subtle phenomena, much of which is still poorly understood. Here these
complexities are studied by formulating the Ginzburg-Landau equations as a
lattice gauge theory. Their solutions are compared and contrasted with the
(heuristic) Landau model of type I superconductivity, and the (perturbative)
Abrikosov model for type II superconductors. Novelties arise as the continuum
limit is approached, related to an effect discovered by Hofstadter. Various
cautionary remarks pertinent to large-scale simulations are made.
