Multigrid Methods Applied to Nonconforming Finite Element Discretizations of the Navier-Stokes Equations

Lutz Tobiska Institut für Analysis und Numerik Otto-von-Guericke-Universität Magdeburg, PF 4120, D-39016 Magdeburg, Germany

Abstract

We consider multigrid methods for nonconforming finite element approximations of the incompressible Navier-Stokes equations

−νΔu + (u · ∇)u + ∇p = f in Ω
∇ · u = 0 in Ω
u = 0 on Γ .

in the twodimensional case. In the first part we summarize some theoretical convergence results for multigrid methods applied to nonconforming finite element discretizations of simplified equations, like the Poisson and the Stokes Problem. In particular we study the nonconforming P1 and the nonconforming rotated Q1 element. In the second part we apply these methods to the Navier-Stokes equations, discuss possibilities for their parallelization and report on some numerical results including the Boussinesq approximation for the nonisothermal Navier-Stokes equations.