A linearized Crank–Nicolson Galerkin FEMs for the nonlinear fractional Ginzburg–Landau equation

ABSTRACT

In this paper, we study a linearized Crank–Nicolson Galerkin finite element method for solving the nonlinear fractional Ginzburg–Landau equation. The boundedness, existence and uniqueness of the numerical solution are studied in details. Then we prove that the optimal error estimates hold unconditionally, in the sense that no restriction on the size of the time step in terms of the spatial mesh size needs to be assumed. Finally, numerical tests are investigated to support our theoretical analysis.

KEYWORDS:

• Fractional Ginzburg–Landau equation
• linearized Crank–Nicolson FEMs
• boundedness
• unique solvability
• convergence
• stability

AMS SUBJECT CLASSIFICATIONS:

• 26A33
• 65M12

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by NSF of China [grant number 11371163]; China Postdoctoral Science Foundation and NSF of Anhui Higher Education Institutions of China [grant number KJ2016A492].