The Crank-Nicolson/Adams-Bashforth Scheme for the Time-Dependent Navier-Stokes Equations with Nonsmooth Initial Data

In this article, we study the stability and convergence of the Crank-Nicolson/Adams-Bashforth scheme for the two-dimensional nonstationary Navier-Stokes equations with a nonsmooth initial data. A finite element method is applied for the spatial approximation of the velocity and pressure. The time discretization is based on the implicit Crank-Nicolson scheme for the linear terms and the explicit Adams-Bashforth scheme for the nonlinear term. Moreover, we prove that the scheme is almost unconditionally stable for a nonsmooth initial data u(0) with divu(0) = 0, i.e., the time step tau satisfies: tau = C(0) if u(0) is an element of H(1) boolean AND L(infinity); tau vertical bar log h vertical bar <= C(0) if u(0) is an element of H(1) for the mesh size h and some positive constant C(0). Finally, we obtain some error estimates for the discrete velocity and pressure under the above stability condition. (C) 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 155-187, 2012