An EMA-balancing, pressure-robust and Re-semi-robust reconstruction method for unsteady incompressible Navier-Stokes equations

Proper EMA-balance (E: kinetic energy; M: momentum; A: angular momentum), pressure-robustness and $Re$-semi-robustness ($Re$: Reynolds number) are three important properties for exactly divergence-free elements in Navier-Stokes simulations. Pressure-robustness means that the velocity error estimates are independent of the pressure approximation errors; $Re$-semi-robustness means that the constants in error estimates do not depend on the inverse of the viscosity explicitly. In this paper, based on the pressure-robust reconstruction method in [Linke and Merdon, ${\it Comput. Methods Appl. Mech. Engrg.}$, 2016], we propose a novel reconstruction method for a class of non-divergence-free simplicial elements which admits all the above properties with only replacing the kinetic energy by a properly redefined discrete energy. We shall refer to it as "EMAPR" reconstruction throughout this paper. Some numerical comparisons with the exactly divergence-free methods, pressure-robust reconstruction methods and methods with EMAC formulation on classical elements are also provided.