  An efficient and accurate finite-element algorithm is described for the
numerical solution of the incompressible Navier-Stokes (INS) equations. The new
algorithm that solves the INS equations in a velocity-pressure reformulation is
based on a split-step scheme in conjunction with the standard finite-element
method. The split-step scheme employed for the temporal discretization of our
algorithm completely separates the pressure updates from the solution of
velocity variables. When the pressure equation is formed explicitly, the
algorithm avoids solving a saddle-point problem; therefore, our algorithm has
more flexibility in choosing finite-element spaces. For efficiency and
robustness, Lagrange finite elements of equal order for both velocity and
pressure are used. Motivated by a post-processing technique that calculates
derivatives of a finite element solution with super-convergent error estimates,
an alternative numerical boundary condition is proposed for the pressure
equation at the discrete level. The new numerical pressure boundary condition
that can be regarded as a better implementation of the compatibility boundary
condition improves the boundary-layer errors of the pressure solution.
Normal-mode analysis is performed using a simplified model problem on a uniform
mesh to demonstrate the numerical properties of our methods. Convergence study
using $\mathbb{P}_1$ elements confirms the analytical results and demonstrates
that our algorithm with the new numerical boundary condition achieves the
optimal second-order accuracy for both velocity and pressure up-to the
boundary. Benchmark problems are also computed and carefully compared with
existing studies. Finally, as an example to illustrate that our approach can be
easily adapted for higher-order finite elements, we solve the classical
flow-past-a-cylinder problem using $\mathbb{P}_n$ finite elements with $n\geq
1$.
