Abstract
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 in the strength of the applied magnetic field with focus on the symmetries of this particular system. The analysis gives new insight in the transitions between stable and unstable states, as well as the connections between stable solution branches.