Assessment of a vorticity based solver for the Navier–Stokes equations

We investigate numerically a recently proposed vorticity based formulation of the incompressible Navier–Stokes equations. The formulation couples a velocity–pressure system with a vorticity–helicity system, and is intended to provide a numerical scheme with enhanced accuracy and superior conservation properties. For a few benchmark problems, we study the performance of a finite element method for this formulation and compare it with the commonly used velocity–pressure based finite element method. It is shown that both steady and unsteady discrete problems in the new formulation admit simple decoupling strategies followed by the application of iterative solves to auxiliary subproblems. Further, we compare several iterative strategies to solve the discrete problems and study the interplay between the choice of stabilization parameters in the finite element method and the efficiency of linear algebra solvers.