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

摘要

We solve the V-V-P, vorticity-velocity-pressure, formulation of the stationary incompressible Navier-Stokes equations based on the least-squares ?nite 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 ?nite elements to enhance the mass conservation of the least-squares method for the in?ow-out?ow problems and to obtain highly accurate results. We demonstrate the advantages of using the higher order ?nite elements and the grid independent solver behavior through the solution of three stationary laminar ?ow problems of benchmarking character. The comparisons show excellent agreement between our results and those of the Galerkin mixed ?nite element method as well as available reference solutions.