Solution of linearized Ginzburg-Landau problem for mesoscopic superconductors by conformal mapping

We present a new method for the solution of linearized Ginzburg-Landau problem for mesoscopic superconducting nanostructures of arbitrary shapes in applied magnetic field. The method is based on the conformal mapping of the analytical solution for the disk and uses a specially designed superconducting gauge for the vector potential corresponding to the magnetic field. As a demonstration of the methods accuracy, we calculate the distribution of the order parameter in superconducting regular polygons and compare the obtained solutions with the available numerical results. We further consider an example of irregular polygon and show the evolution of the vortex patterns in function of the geometry of samples boundary. The obtained results will be compared with available experimental data on mesoscopic and nanoscopic superconductors.