A Multigrid Method for Incompressible Flow Problems Using Quasi Divergence Free Functions

We consider a finite element method of higher order on a quadrilateral or triangular, hexahedral or tetrahedral mesh for solving the Stokes or Navier–Stokes problem by means of discontinuous elements for the pressure and suitable conforming elements for the velocity such that the global and the local inf-sup condition are satisfied. Our goal is the construction of a multigrid solver for the corresponding algebraic system of equations. The idea of this solver is to switch inside of the multigrid method to a new velocity basis which leads to a reduced system with a much lower number of unknowns as well as a very small number of couplings between them. We call this new basis quasi divergence free since most of the basis functions are dicretely divergence free which implies that they do not have any coupling to the pressure. The quasi divergence free basis functions can be constructed locally during the assembling process of the stiffness matrix. We create a multigrid method for solving the reduced problem efficiently. Since most of the velocity basis functions are completely decoupled from the pressure we can construct a smoother with low computational costs. The efficiency of the new multigrid method compared with other known multigrid solvers is demonstrated by numerical experiments for the Stokes problem. It is shown how the ideas for the construction of a quasi divergence free basis can be extended from the Stokes equations to the incompressible Navier-Stokes equations in the stationary and non-stationary case.