Numerical analysis of a linear second-order energy-stable auxiliary variable method for the incompressible Navier–Stokes equations

In this paper, the unconditional stability and convergence analysis of a second-order linear finite element scheme based on the auxiliary variable method are studied for the incompressible time-dependent Navier–Stokes equations. Firstly, a corresponding equivalent system of the Navier–Stokes equations with three variables is formulated by introducing a nonlocal variable and designing an additional ordinary differential equation for it which plays the key role to maintain the unconditional energy stability. Secondly, a fully discrete scheme is developed and the stable finite element spaces are adopted to approximate the spatial variables, which is implicit for the linear terms and explicit for the nonlinear term. Hence, one only needs to solve several constant coefficient algebraic systems at each time step illustrating the high practical efficiency. Numerical experiments are presented to verify the theoretical results.