A linearized BDF2 virtual element method for the unsteady Brinkman–Forchheimer equations with variable time step

Abstract

This paper is devoted to a lowest order virtual element method for the unsteady incompressible Brinkman–Forchheimer equations on polygonal mesh, while a linearized variable-time-step second order backward differentiation formula is adopted in time. We utilize discrete orthogonal convolution and discrete complementary convolution kernels to obtain error estimates for the solutions of time-discrete system. By virtue of the temporal–spatial error splitting approach, and under the current mildest adjacent time-step ratio condition: 0<rk≔τk/τk−1≤rmax≈4.8645, we have established the L∞ boundedness of the fully discrete velocity solution and its L2-norm optimal error estimate. Numerical experiments are conducted on various polygonal meshes, validating the accuracy of the theoretical analysis.