Optimal error estimates of linearized Crank-Nicolson Galerkin FEMs for the time-dependent Ginzburg-Landau equations in superconductivity

In this paper, we study linearized Crank-Nicolson Galerkin finite element methods for time-dependent Ginzburg-Landau equations under the Lorentz gauge. We present an optimal error estimate for the linearized schemes (almost) unconditionally (i.e., when the spatial mesh size h and the temporal step τ are smaller than a given constant), while previous analyses were given only for some schemes with strong restrictions on the time step-size. The key to our analysis is the boundedness of the numerical solution in some strong norm. We prove the boundedness for the cases τ ≥ h and τ ≤ h, respectively. The former is obtained by a simple inequality, with which the error functions at a given time level are bounded in terms of their average at two consecutive time levels, and the latter follows a traditional way with the induction/inverse inequality. Two numerical examples are investigated to confirm our theoretical analysis and to show clearly that no time step condition is needed.