A linearized Crank-Nicolson-Galerkin FEM for the time-dependent Ginzburg-Landau equations under the temporal gauge (Q2875716)

The author proposes a decoupled and linearized fully discrete finite element method (FEM) for the time-dependent Ginzburg-Landau equations under the temporal gauge, where a Crank-Nicolson scheme is used for the time discretization. By carefully designing the time-discretization scheme, the author controls to prove the convergence rate \({\mathcal O}(\tau^2 + h^r)\), where \(\tau\) is the time-step size and \(r\) is the degree of the finite element space. Due to the degeneracy of the problem, the convergence rate in the spatial direction is one order lower than the optimal convergence rate of FEMs for parabolic equations. Numerical tests are presented that support the theoretical conclusion.