A new error analysis of backward Euler Galerkin finite element method for two-dimensional time-dependent Ginzburg–Landau equation

Abstract

In this paper, the backward Euler Galerkin finite element method (FEM) is investigated for the two dimensional time-dependent Ginzburg–Landau equation. The unconditionally optimal L2 error estimate is obtained without using the boundedness of the numerical solution in L∞ norm, while it is an indispensable requirement in the previous works. A key to the analysis is to deal with the nonlinear term rigorously and skillfully in terms of the boundedness of the numerical solution in L2 norm rather than the L∞ norm. Numerical results are presented to confirm the theoretical findings.