Analysis and computation of a pressure-robust method for the rotation form of the stationary incompressible Navier-Stokes equations by using high-order finite elements

TL;DR

This paper introduces a high-order, pressure-robust finite element method for the rotation form of stationary incompressible Navier-Stokes equations, improving accuracy and stability through velocity reconstruction and novel discretization techniques.

Contribution

It develops a new pressure-robust discretization using an H(div)-conforming reconstruction operator and a skew-symmetric trilinear form, with proven optimal convergence rates.

Findings

Achieves pressure-independent velocity errors

Demonstrates optimal convergence rates for velocity and pressure

Numerical experiments confirm theoretical advantages

Abstract

In this work, we develop a high-order pressure-robust method for the rotation form of the stationary incompressible Navier-Stokes equations. The original idea is to change the velocity test functions in the discretization of trilinear and right hand side terms by using an H(div)-conforming velocity reconstruction operator. In order to match the rotation form and error analysis, a novel skew-symmetric discrete trilinear form containing the reconstruction operator is proposed, in which not only the velocity test function is changed. The corresponding well-posed discrete weak formulation stems straight from the classical inf-sup stable mixed conforming high-order finite elements, and it is proven to achieve the pressure-independent velocity errors. Optimal convergence rates of H1, L2-error for the velocity and L2-error for the Bernoulli pressure are completely established. Adequate numerical experiments are presented to demonstrate the theoretical results and the remarkable performance of the proposed method.