A split-step finite-element method for incompressible Navier-Stokes equations with high-order accuracy up-to the boundary

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. In contrast, popular mixed finite-element methods that solve the INS equations in the primitive variables (or velocity-divergence formulation) lead to discrete saddle-point problems whose solution depends on the choice of finite-element spaces for velocity and pressure that is subject to the well-known Ladyzenskaja-BabuSla-Brezzi (LBB) (or inf-sup) condition. For efficiency and robustness, Lagrange (piecewise-polynomial) finite elements of equal order for both velocity and pressure are used. Accurate numerical boundary condition for the pressure equation is also investigated. 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. A normal-mode analysis is performed using a simplified model problem on a uniform mesh to demonstrate the numerical properties of our methods. Convergence studies using P-1 elements support 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 P-n finite elements with n >= 1. (C) 2020 Elsevier Inc. All rights reserved.