Analysis of a Crank-Nicolson fast element-free Galerkin method for the nonlinear complex Ginzburg-Landau equation

A fast element-free Galerkin (EFG) method is proposed in this paper for solving the nonlinear complex Ginzburg-Landau equation. A second-order accurate time semi-discrete system is presented by using the Crank-Nicolson scheme for the temporal discretization, and then a meshless fully discrete system is established by using the EFG method for the spatial discretization. In the proposed EFG method, Nitsche's technique is used to impose the essential boundary conditions in a weak sense, and the reproducing kernel gradient smoothing integration is used to accelerate the calculation. Theoretical errors for the time semi-discrete system and the fully discrete EFG system are analyzed in detail. Optimal error estimates of the fully discrete Crank-Nicolson EFG method are obtained in both L 2 and H 1 norms. Numerical results validate the theoretical results and the effectiveness of the method.