Quantized vortex, a well-known signature of superfluidity, was observed in superconductors half a century ago, and more recently, it has also been observed in the Bose-Einstein condensates, the so-called fifth state of matter.
In this talk, we will present some numerical algorithms for the computation of vortices in superconductors based on the phenomenological Ginzburg-Landau models and also vortices in the Bose-Einstein condensate in a rotating trap based on the Gross-Pitaevskii equations. Some striking similarities will be illustrated. A brief introduction to the physical background of these problems will be given. Some mathematical results concerning these equations as well as the computational challenges will be discussed. Numerical simulations and their physical interpretations will also be presented.