An extension of quasiclassical Keldysh-Usadel theory to higher spatial
dimensions than one is crucial in order to describe physical phenomena like
charge/spin Hall effects and topological excitations like vortices and
skyrmions, none of which are captured in one-dimensional models. We here
present a numerical finite element method which solves the non-linearized 2D
and 3D quasiclassical Usadel equation relevant for the diffusive regime. We
show the application of this on three model systems with non-trivial
geometries: (i) a bottlenecked Josephson junction with external flux, (ii) a
nanodisk ferromagnet deposited on top of a superconductor and (iii)
superconducting islands in contact with a ferromagnet. In case (i), we
demonstrate that one may control externally not only the geometrical array in
which superconducting vortices arrange themselves, but also to cause
coalescence and tune the number of vortices. In case (iii), we show that the
supercurrent path can be tailored by incorporating magnetic elements in planar
Josephson junctions which also lead to a strong modulation of the density of
states. The finite element method presented herein paves the way for gaining
insight in physical phenomena which have remained largely unexplored due to the
complexity of solving the full quasiclassical equations in higher dimensions.Comment: 16 pages, 8 figures. Added several new result
