Abstract We study the vortex structures and quasi-particle structures in nano-structured superconductors. We used the Bogoliubov–de Gennes equation and the finite element method and obtained stable magnetic flux structures and the quasi-particle states. We found the vortex configurations are affected by the interference of the quasi-particle bound states around the vortices. In order to clarify the interference between the quasi-particle wave-functions around two vortices we have developed a numerical method using the elliptic coordinates and the Mathieu functions. We apply this method to two singly quantized vortex state in a conventional s -wave superconductor and a pair of half-quantum vortices in a chiral p -wave superconductor.