Two-Level Spline Approximations for Two-Dimensional Navier-Stokes Equations

The C-1 spline spaces with degree d >= 5 over given triangulations are implemented in the framework of multi-variate spline theory. Based on this approach, two-level methods are proposed by using various order spline spaces for the steady state Navier-Stokes equations in the stream function formulation. The proposed method can be reduced to solving a linear equation in the high-order spline space and the nonlinear equations in the low-order spline space. The convergence analysis is given based on the Newton iteration. Besides, the matrix forms of the two-level scheme are also presented. We finally tabulate the numerical results to validate and show the efficiency of the proposed two-level spline methods.