Convergence and superconvergence analysis of finite element methods for nonlinear Ginzburg–Landau equation with Caputo derivative

In this paper, we study and analyze the time fractional Ginzburg–Landau equation (FGLE) using finite element methods (FEMs) in space and L 1 scheme in time. The unconditional optimal L^2 -norm error estimates are obtained based on the time-space error splitting technique. Using the relation between interpolation operator and Ritz projection operator, we obtain the superclose results in H^1 -norm. Furthermore, the global superconvergence results are established through the interpolation postprocessing technique. To overcome the weak singularity of the solution at the initial time and improve the computational efficiency, we adopt the nonuniform L 1 scheme in the time direction and built corresponding fast algorithm based on sum-of-exponential technique. Finally, we provide several numerical experiments to verify the theoretical results and demonstrate the advantages of the fast algorithm.