Schrödinger eigenbasis on a class of superconducting surfaces: Ansatz, analysis, FEM approximations and computations

In this work we focus on the efficient representation and computation of the eigenvalues and eigenfunctions of the surface Schrödinger operator (i∇+A0)2 that governs a class of nonlinear Ginzburg-Landau (GL) superconductivity models on rotationally symmetric Riemannian 2-manifolds S. We identify and analyze a complete orthonormal system in L2(S;C) of eigenmodes having a variable-separated form. For the unknown functions in this ansatz, our analysis facilitates the identification of approximate spectral problems whose eigenvalues lie arbitrarily near corresponding eigenvalues of the Schrödinger operator. We then develop and implement an arbitrary order finite element method for the efficient numerical approximation of the eigenvalue problem. We also demonstrate our analysis, algorithm and its convergence rate using parallel computations performed on a variety of choices of smooth and non-smooth surfaces S.