A linearized element-free Galerkin method for the complex Ginzburg-Landau equation

In this paper, an effective linearized element-free Galerkin (EFG) method is developed for the numerical solution of the complex Ginzburg-Landau (GL) equation. To deal with the time derivative and the nonlinear term of the GL equation, an explicit linearized procedure is presented. The unconditional stability and the error estimate of the procedure are analyzed. Then, a stabilized EFG method is proposed to establish linear algebraic systems. In the method, the penalty technique is used to facilitate the satisfaction of boundary conditions, and the stabilized moving least squares approximation is used to enhance the stability and performance. The linearized EFG method is a meshless method and possesses high precision and convergence rate in both space and time. Theoretical error and convergence of the linearized EFG method are analyzed. Finally, some numerical results are provided to demonstrate the efficiency of the method and confirm the theoretical results.

Automated summary

This paper presents an effective linearized element-free Galerkin (EFG) method for solving the complex Ginzburg-Landau (GL) equation, with high precision and convergence rate. A stabilized moving least squares approximation is proposed to enhance stability and performance, and a penalty technique is used to satisfy boundary conditions. Numerical results demonstrate the efficiency of the method.