  Thin film magnetization problems in type-II superconductivity are usually
formulated in terms of the magnetization function alone, which allows one to
compute the sheet current density and the magnetic field but often inhibits
computing the electric field in the film. Accounting for the current leads
presents an additional difficulty encountered in thin film transport current
problems.
  We generalize, to the presence of a transport current, the two-variable
variational formulation proposed recently for thin film magnetization problems.
The formulation, written in terms of the magnetization function and the
electric field, is used as a basis for a new numerical approximation enabling
us to solve the magnetization and transport current problems for flat films of
arbitrary shapes, including multiply connected films. The advantage of this
approach is in its ability to compute accurately all variables of interest,
including the electric field, for any value of the power in the power law
current-voltage relation characterizing the superconducting material. In the
high power limit the critical state model solution is obtained.
