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

## TL;DR

This paper introduces a split-step finite-element method for incompressible Navier-Stokes equations that achieves high-order accuracy up to the boundary, improving efficiency and boundary-layer error handling.

## Contribution

The paper presents a novel split-step finite-element algorithm with a new pressure boundary condition that enhances accuracy and flexibility for solving incompressible Navier-Stokes equations.

## Key findings

- Achieves second-order accuracy for velocity and pressure up to the boundary.
- Demonstrates robustness and efficiency with equal-order Lagrange finite elements.
- Successfully applied to flow past a cylinder with higher-order elements.

## Abstract

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 $n\geq 1$.