Abstract: 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 superconducting material with support of the fields contained primarily in O(λL) 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 λL on complex three dimensional geometries. However, for typical materials λL∼O(1e−7), which makes the PDE and integral equation increasingly difficult to solve in the limit λL→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(λL) accurate as compared to the corresponding solutions of the London equations and the Debye source integral equations respectively. We demonstrate the effectiveness of this asymptotic approach both in terms of speed and accuracy through several numerical examples.
Bio: Manas Rachh joined the Simons foundation as part of the Numerical Algorithms group at Flatiron’s Center for Computational Biology in 2018, and is currently a research scientist in the Center for Computational Mathematics. His research interests include partial differential equations (PDEs) arising in mathematical physics, integral equation methods, robust computation of eigenvalues and eigenfunctions of elliptic PDEs, and the development of fast algorithms for applications in electrostatics, acoustics, viscous flow, electromagnetics, biomedical imaging, and data visualization. Before coming to the foundation, he obtained his Ph.D. from the Courant Institute of Mathematical Sciences at New York University with Leslie Greengard followed by a Gibbs Assistant Professorship at Yale University where he worked with Vladimir Rokhlin.