On first-order formulations of the least-squares finite element method for incompressible flows

To reduce the element continuity requirement for the least-squares finite element method (LSFEM), it is customary to rewrite the Navier-Stokes equations as first-order partial differential equations. In this work, numerical experiments for three-dimensional incompressible flows are carried out by using the LSFEM based on three types of first-order formulations, namely the velocity-vorticity-pressure (VVP) formulation, the velocity-stress-pressure (VSP) formulation and the velocity-velocity gradient-pressure (VVGP) formulation. In addition, two modifications for each formulation are considered. Numerical results indicate that proper problem formulation can significantly reduce computing time and improve the accuracy of numerical solutions.