Manas Rachh, Center for Computational Mathematics, Flatiron Institute
Static Currents in Type-I superconductors
In this talk, we describe the classical magneto-static approach to the theory of type-I superconductors. The magnetic field and the current in type-I superconductors are related by the London equations and tend to decay exponentially inside the supercon-ducting material with support of the fields contained primarily in $O(\lambda L)$ neighborhood of the superconductor. We present a Debye source based integral representation for the numerical solution of the London equations, and demonstrate the efficacy of our approach for moderate values of $\lambda L$ on complex three dimensional geometries. How-ever, for typical materials $\lambda L \sim O(10^{-7})$, which makes the PDE and integral equation increasingly difficult to solve in the limit $\lambda L \to 0$ due to the presence of two different length scales in the problem. We derive a limiting PDE and a corresponding integral equation, and show that the solutions of this limiting PDE and integral equations are $O(\lambda L)$ accurate as compared to the corresponding solutions of the London equations and the Debye source integral equations respectively. We demonstrate the effective-ness of this asymptotic approach both in terms of speed and accuracy through several numerical examples.