On Unconditional Stability of a Variable Time Step Scheme for the Incompressible Navier-Stokes Equations

In this work, an unconditionally stable, decoupled, variable time step scheme is presented for the incompressible Navier-Stokes equations. Based on a scalar auxiliary variable in exponential function, this fully discrete scheme combines the backward Euler scheme for temporal discretization with variable time step and a mixed finite element method for spatial discretization, where the nonlinear term is treated explicitly. Moreover, without any restriction on the time step, stability of the proposed scheme is discussed. Besides, error estimate is provided. Finally, some numerical results are presented to illustrate the performances of the considered numerical scheme.