Convergence of linearized backward Euler-Galerkin finite element methods for the time-dependent Ginzburg-Landau equations with temporal gauge

摘要

We present error analysis of fully discrete Galerkin finite element methods for the time-dependent Ginzburg-Landau equations with the temporal gauge, where a linearized backward Euler scheme is used for the time discretization. We prove that the convergence rate is O(τ + h~r) if the finite element space of piecewise polynomials of degree r is used. Due to the degeneracy of the problem, the convergence rate is one order lower than the optimal convergence rate of finite element methods for parabolic equations. Numerical examples are provided to support our theoretical analysis.