Multiscale and hysteresis effects in vortex pattern simulations for Ginzburg–Landau problems

Abstract

SUMMARYThe behavior and properties of vortex pattern solutions to a benchmark Ginzburg-Landau model are investigated using hybrid continuation algorithms in conjunction with parallel adaptive mesh refinement (AMR) schemes to resolve the local vortices. The model is related to phase transition models arising in superconductivity and superfluids. The approach is based on a coupled variational formulation and finite element approximation scheme for the complex-valued solution. The associated algorithms implement continuation treatments based on the vortex scale coherence parameter and the winding number parameter in this model. Simulation results demonstrate the behavior of non-unique solutions, as characterized by different vortex configurations, and energy plots are used to display hysteresis effects. The complex-valued nature of the solution also serves to illustrate some interesting open questions related to AMR strategies and error indicators for complex-valued solution fields as well as other implications for such coupled systems.

An optimal linear solver for the Jacobian system of the extreme type-II Ginzburg–Landau problem

“…As the matrix that belongs to edge between k 1 , k 2 is the only matrix with a nonzero entry at (k 1 , k 2 ) (namely −1), its coefficient in ( 12) must be 0. Similarly, the same holds for all other coefficients, such that (11) can only be fulfilled of β i = 0 for all i ∈ {1, . .…”

“…Also, preconditioned Newton-Krylov methods are already applied in other phase field models such as the Cahn-Hilliard equation [10]. Initial efforts to apply Newton-Krylov to the Ginzburg-Landau problem were taken in [11]; preconditioning is not discussed though.…”