Unconditional optimal error estimates of a modified finite element fully discrete scheme for the complex Ginzburg-Landau equation

In this paper, based on the 2-step backward differentiation formula (BDF2 for short) in time and the nonconforming Wilson element in space, a modified Galerkin finite element method named BDF2-MG FEM is proposed to solve the nonlinear complex Ginzburg-Landau equation (GLE for short). On one hand, a modified Ritz projection operator R-h is introduced and analyzed, which plays an important role in getting the unconditional optimal error estimates. On the other hand, a time-discrete system is constructed with the linearized BDF2 and the regularity is derived with the temporal error results. Combining these two aspects, the errors between RhUn and U-h(n) with order O(h(3)+h(2)Delta t) in L-2-norm and O(h(2)+h(2)Delta t) in the modified energy norm are deduced, where h is the subdivision parameter, Delta t is the time step, U-n and U-h(n) denote the solutions of the time-discrete system and the BDF2-MG FEM respectively. Therefore the boundedness of parallel to U-h(n)parallel to(0,infinity) is proven without any restriction on the time-space grid ratio. Furthermore, by using the properties of R-h, unconditional optimal error estimates of order O(h(3)+(Delta t)(2)) in L-2-norm and O(h(2)+(Delta t)(2)) in the modified energy norm are obtained directly. It should point out the spatial discrete errors of the BDF2-MG FEM are all one order higher than that of the BDF2 traditional Galerkin finite element method with Wilson element for the GLE. At last, a numerical experiment is presented to verify the validity of the theoretical analysis.

Automated summary

In this paper, a modified Galerkin finite element method named BDF2-MG FEM is proposed to solve the nonlinear complex Ginzburg-Landau equation (GLE) based on the 2-step backward differentiation formula (BDF2) in time and the nonconforming Wilson element in space. The theoretical analysis shows that the method has unconditional optimal error estimates. A numerical experiment is presented to verify the validity of the proposed method.