A numerical experiment on the stability and convergence characteristics of some splitting mixed-finite element methods for solving the incompressible Navier-Stokes equations

In the present study, a fractional three-step P2P1 finite element method (FEM) for solving the unsteady incompressible Navier-Stokes equations, which is a variation of P1P1 four-step splitting FEM [1], was compared with conventional one-step time-integration schemes in terms of the CPU time and convergence characteristics of an iterative solver by the solution of some benchmark problems. One-step time-integration schemes were temporarily discretized by either the Crank-Nicolson or the Adams-Bashforth method. Fractional three-step P2P1 FEM consists of three steps: a non-linear momentum equation with the pressure in the previous time step is solved to obtain an intermediate velocity field by the Crank-Nicolson method in the first step and another intermediate velocity field is calculated using the pressure in the previous time step in the second step, and a divergence-free constraint is imposed in the last step to update the pressure field, in which a symmetric saddle-point type matrix (SPTM) is solved. It was shown that the fractional three-step method is more efficient than one-step time-integration schemes because a symmetric SPTM is assembled only once during the entire computation and solved once at each time-step; further, the cost of solving the nonlinear momentum equation in a fully-implicit manner is relatively low. Furthermore, the proposed method was found to be more stable than one-step time-integration schemes as it provided stable solutions at higher CFL numbers.