Finite element method for the quasiclassical theory of superconductivity

The Eilenberger-Larkin-Ovchinnikov-Eliashberg quasiclassical theory of superconductivity is a powerful method enabling studies of a wide range of equilibrium and nonequilibrium phenomena in conventional and unconventional superconductors. We introduce here a finite element method, based on a discontinuous Galerkin approach, to self-consistently solve the underlying transport equations for general device geometries, arbitrary mean free path, and symmetry of the superconducting order parameter. We present results on (i) the influence of scalar impurity scattering on phase crystals in $d$-wave superconducting grains at low temperatures and (ii) the current flow and focusing in $d$-wave superconducting weak links, modeling recent experimental realizations of grooved high-temperature superconducting Dayem bridges. The high adaptability of this finite element method for quasiclassical theory paves the way for future investigations of superconducting devices and new physical phenomena in unconventional superconductors.