A posteriori error estimates and time adaptivity for fully discrete finite element method for the incompressible Navier-Stokes equations

In this paper, we study a posteriori error estimates for the incompressible Navier-Stokes equations in a convex polygonal domain. The semi-implicit variable step-size two-step backward differentiation formula (BDF2) is employed for the time discretization and the Taylor–Hood finite element method (FEM) is used for the space discretization. We prove energy stability of semi-implicit variable step-size BDF2 FEM under different Courant Friedreich Lewy (CFL)-type conditions by utilizing different embeddings for the nonlinear term. Two appropriate reconstructions of the approximate solution are proposed to obtain the time discretization error. Resorting to the energy stability and the quadratic reconstructions, we obtain a posteriori upper and lower error bounds for the fully discrete approximation. We further develop a time adaptive algorithm for efficient time step control based on the time error estimators. Several numerical experiments are performed to verify our theoretical results and demonstrate the efficiency of the time adaptive algorithm.