UNCONDITIONAL CONVERGENCE OF HIGH-ORDER EXTRAPOLATIONS OF THE CRANK-NICOLSON, FINITE ELEMENT METHOD FOR THE NAVIER-STOKES EQUATIONS

作者：Ingram Ross^*

来源：International Journal of Numerical Analysis and Modeling, 2013, 10(2): 257-297.

• Navier-Stokes
• Crank-Nicolson
• finite element
• extrapolation
• linearization
• error
• convergence
• linearization

Error estimates for the Crank-Nicolson in time, Finite Element in space (CNFE) discretization of the Navier-Stokes equations require application of the discrete Gronwall inequality, which leads to a time-step (Delta t) restriction. All known convergence analyses of the fully discrete CNFE with linear extrapolation rely on a similar Delta t-restriction. We show that CNFE with arbitrary-order extrapolation (denoted CNLE) is convergences optimally in the energy norm without any Delta t-restriction. We prove that CNLE velocity and corresponding discrete time-derivative converge optimally in l(infinity) (H-1) and l(2)(L-2) respectively under the mild condition Delta t <= Mh(1/4) for any arbitrary M > 0 (e.g. independent of problem data, h, and Delta t) where h > 0 is the maximum mesh element diameter. Convergence in these higher order norms is needed to prove convergence estimates for pressure and the drag/lift force a fluid exerts on an obstacle. Our analysis exploits the extrapolated convective velocity to avoid any Delta t-restriction for convergence in the energy norm. However, the coupling between the extrapolated convecting velocity of usual CNLE and the a priori control of average velocities (characteristic of CN methods) rather than pointwise velocities (e.g. backward-Euler methods) in l(2)(H-1) is precisely the source of Delta t-restriction for convergence in higher-order norms.

• 出版日期2013