An efficient iterative method for dynamical Ginzburg-Landau equations

TL;DR

This paper introduces a finite element method with an efficient preconditioner for simulating time-dependent Ginzburg-Landau equations, improving stability and computational speed especially for complex geometries.

Contribution

It presents a novel finite element approach using Nedelec elements and a new preconditioner for the Newton iteration, enhancing simulation efficiency and stability.

Findings

Preconditioner significantly speeds up large-scale simulations.

Method effectively handles boundary conditions and complex geometries.

Numerical results confirm the approach's efficiency and stability.

Abstract

In this paper, we propose a new finite element approach to simulate the time-dependent Ginzburg-Landau equations under the temporal gauge, and design an efficient preconditioner for the Newton iteration of the resulting discrete system. The new approach solves the magnetic potential in H(curl) space by the lowest order of the second kind Nedelec element. This approach offers a simple way to deal with the boundary condition, and leads to a stable and reliable performance when dealing with the superconductor with reentrant corners. The comparison in numerical simulations verifies the efficiency of the proposed preconditioner, which can significantly speed up the simulation in large-scale computations.