Applied Math Seminar

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Static Currents in Type-I Superconductors

Speaker: Manas Rachh, Center for Computational Mathematics, Flatiron Institute

Location: Warren Weaver Hall 1302

Date: Friday, March 14, 2025, 2:30 p.m.

Synopsis:

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 \(\mathcal O(\lambda L)\) 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 \(\lambda L \sim \mathcal O(10^{-7})\), 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
\(\mathcal O(\lambda L)\) 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.