Newton multigrid least-squares FEM for the V-V-P formulation of the Navier–Stokes equations

We solve the V-V-P, vorticity–velocity–pressure, formulation of the stationary incompressible Navier–Stokes equations based on the least-squares finite element method. For the discrete systems, we use a conjugate gradient (CG) solver accelerated with a geometric multigrid preconditioner for the complete system. In addition, we employ a Krylov space smoother inside of the multigrid which allows a parameter-free smoothing. Combining this linear solver with the Newton linearization, we construct a very robust and efficient solver. We use biquadratic finite elements to enhance the mass conservation of the least-squares method for the inflow–outflow problems and to obtain highly accurate results. We demonstrate the advantages of using the higher order finite elements and the grid independent solver behavior through the solution of three stationary laminar flow problems of benchmarking character. The comparisons show excellent agreement between our results and those of the Galerkin mixed finite element method as well as available reference solutions.