Energy stability and error estimates of a maximum bound principle preserving scheme for the dynamic Ginzburg-Landau equations of superconductivity

The paper proposes a decoupled numerical scheme of the time-dependent Ginzburg-Landau equations under temporal gauge. For the order parameter and the magnetic potential, the discrete scheme adopts the second type Ned${\rm \acute{e}}$lec element and the linear element for spatial discretization, respectively, and a fully linearized backward Euler method and the first order exponential time differencing method for time discretization, respectively. The maximum bound principle of the order parameter and the energy dissipation law in the discrete sense are proved for this finite element-based scheme. This allows the application of the adaptive time stepping method which can significantly speed up long-time simulations compared to existing numerical schemes, especially for superconductors with complicated shapes. The error estimate is rigorously established in the fully discrete sense. Numerical examples verify the theoretical results of the proposed scheme and demonstrate the vortex motions of superconductors in an external magnetic field.