Workshop on Numerical Methods and Analysis in CFD - Abstract

Li, Xu

An EMA-conserving, pressure-robust and Re-semi-robust reconstruction method for incompressible Navier--Stokes simulations

Proper EMA-balance (E: kinetic energy; M: momentum; A: angular momentum), pressure-robustness and Re-semi-robustness (Re: Reynolds number) are three important properties of Navier--Stokes simulations with exactly divergence-free elements. This EMA-balance makes a method conserve kinetic energy, linear momentum and angular momentum under some suitable senses; pressure-robustness means that the velocity errors are independent of the continuous pressure; Re-semi-robustness means that the constants appearing in the error bounds of kinetic and dissipation energies do not explicitly depend on inverse powers of the viscosity. In this paper, based on the pressure-robust methods in [1], we propose a novel reconstruction method for a class of non-divergence-free simplicial elements which admits almost all the above properties. The only exception is the energy balance, where kinetic energy should be replaced by a properly redefined discrete energy. Our lowest order case uses the Bernardi-Raugel element on general shape-regular meshes. Some numerical comparisons with exactly divergence-free methods, pressure-robust reconstructions and the EMAC scheme are provided to confirm our theoretical results.

References:
[1] A. Linke and C. Merdon, Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 311 (2016), pp. 304--326.