Unconditional optimal [formula omitted]-norm error estimate and superconvergence analysis of a linearized nonconforming finite element variable-time-step BDF2 method for the nonlinear complex Ginzburg–Landau equation

Abstract

In this paper, a linearized fully discrete scheme combining the variable-time-step two-step backward differentiation formula (VSBDF2) in time and the nonconforming finite element methods (FEMs) in space is constructed and analyzed for the nonlinear complex Ginzburg–Landau equation. A novel convergence analysis approach is proposed, which shows that the H1-norm error of this scheme can reach optimal order in space and sharp second-order convergence in time. The key ingredients of our analysis are temporal–spatial error splitting approach, a new positive definiteness property of discrete orthogonal convolution (DOC) kernels with respect to complex sequences, an energy projection operator and a discrete Laplace operator Δh related to the nonconforming element space, and a relationship between Δh and classical L2 projection operator in complex space. Furthermore, through a specific treatment of the error equation and the combination technique of interpolating operator and projection operator, we prove that the proposed scheme has the global superconvergence result of order O(τ2+h2) in H1-norm for the first time, which further improves the efficiency of numerical computation. Here τ and h are the maximum time and spatial step sizes, respectively. It should be pointed out that all these error results are rigorously established under mild assumptions of the time step, and get rid of the usual grid ratio constraints between temporal and spatial step sizes in linearized scheme. At last, numerical experiments are provided to verify the theoretical analysis.