Unconditional error estimate of a linearized variable-time-step BDF2 nonconforming virtual element method for nonlinear coupled predator–prey equations

In this paper, we introduce the Nonconforming Virtual Element Method (NVEM) to solve nonlinear coupled prey–predator equations on general polygonal meshes, employing a linearized variable two-step backward differentiation formula (BDF2) for time discretization. The analysis of boundedness and error estimates for the time discrete system is conducted using discrete orthogonal convolution kernels and discrete complementary convolution kernels. Then, the time–space error splitting technique and the projection operator are integrated to establish the L∞-norm boundedness of the fully discrete solution, independent of any grid ratio conditions, thereby naturally deriving the unconditionally optimal error estimate. The analytical method presented herein is not restricted to the NVEM and can be readily extended to other numerical techniques. Finally, the theoretical results are validated through numerical examples, demonstrating the scheme’s effectiveness across various grid configurations.