  This paper considers the extreme type-II Ginzburg-Landau equations that model
vortex patterns in superconductors. The nonlinear PDEs are solved using
Newton's method, and properties of the Jacobian operator are highlighted.
Specifically, it is illustrated how the operator can be regularized using an
appropriate phase condition. For a two-dimensional square sample, the numerical
results are based on a finite-difference discretization with link variables
that preserves the gauge invariance. For two exemplary sample sizes, a thorough
bifurcation analysis is performed using the strength of the applied magnetic
field as a bifurcation parameter and focusing on the symmetries of this system.
The analysis gives new insight in the transitions between stable and unstable
states, as well as the connections between stable solution branches.
