Unconditional Error Estimate of a Nonconforming Virtual Element Method for Allen-Cahn Equation with a Variable-Time-Step BDF2 Scheme

A nonconforming virtual element scheme for the Allen-Cahn equation on polygonal meshes is developed. The numerical scheme employs a variable-time-step two-step backward differentiation formula (BDF2) for temporal discretization and a lowest-order nonconforming virtual element method in space. With the help of discrete orthogonal convolution (DOC) and discrete complementary convolution (DCC) kernels, we obtain the error estimates of temporal semi-discrete system under a mild time-step ratio $0<r_k<r_{max}≈4.8645.$ By using temporal-spatial error splitting approach, we establish the L^1-norm bound of discrete solution, and the unconditional optimal $L^2-$error estimates are carried out without CFL condition. Numerical experiments validate the theoretical findings across diverse polygonal meshes.