Superconductors are described by minimizers of the Ginzburg-Landau energy, which can exhibit fascinating macroscopic phenomena such as the formation of Abrikosov vortex lattices under an external magnetic field. The numerical approximation of these states is challenging due to resolution conditions on the computational mesh, which can be expressed mathematically by error bounds of the discrete minimizers involving the mesh size and a material parameter in the Ginzburg-Landau energy functional. In this talk, I present recent advances that address these challenges using approximation spaces constructed via the Localized Orthogonal Decomposition - a multiscale technique that leads to significant relaxation of the mesh resolution condition. These developments enable more efficient and accurate numerical approximations of vortex lattice structures in superconductors, thereby offering new opportunities for the computational exploration of complex phenomena in the Ginzburg?Landau model.