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In this paper, we study the stability and convergence of the Crank-Nicolson/AdamsBashforth scheme for the two-dimensional nonstationary Navier-Stokes equations. A finite element method is applied for the spatial approximation of the velocity and pressure. The time discretization is based on the Crank-Nicolson scheme for the linear term and the explicit Adams-Bashforth scheme for the nonlinear term. Moreover, we present optimal error estimates and prove that the scheme is almost unconditionally stable and convergent, i.e., stable and convergent when the time step is less than or equal to a constant.

In this paper, we study the unconditional convergence and error estimates of a Galerkin-mixed FEM with the linearized semi-implicit Euler time-discrete scheme for the equations of incompressible miscible flow in porous media. We prove that the optimal L 2 error estimates hold without any time-step (convergence) conditions, while all previous works require certain time-step restrictions. Our theoretical results provide a new understanding on commonly-used linearized schemes. The proof is based on a splitting of the error into two parts: the error from the time discretization of the PDEs and the error from the finite element discretization of corresponding time-discrete PDEs. The approach used in this paper can be applied to more general nonlinear parabolic systems and many other linearized (semi)-implicit time discretizations.