Unconditional error analysis of a linearized BDF2 virtual element method for nonlinear Ginzburg–Landau equation with variable time step

Abstract

We consider a virtual element method in space for the nonlinear Ginzburg–Landau equation, while a linearized time-variable-step second order backward differentiation formula (BDF2) is adopted in time. The error splitting approach is used to prove the unconditional optimal error estimate of the derived scheme under the mild restriction on the ratio of adjacent time-steps ratios (similarly proposed in Liao and Zhang 2021, Zhang and Zhao 2021). By using the techniques of the discrete complementary convolution (DOC) kernels and the discrete complementary convolution (DCC) kernels, we obtain the boundedness and error estimates of the solution of time-discrete system. Moreover, the optimal convergence in L2-norm for the fully discrete scheme is finally derived. Numerical examples on a set of polygonal meshes are given to validate our theoretical results.