Efficient Newton-multigrid solution techniques for higher order space-time Galerkin discretizations of incompressible flowI

In this paper, we discuss solution techniques of Newton-multigrid type for the resulting nonlinear saddle-point block-systems if higher order continuous Galerkin-Petrov (cGP(k)) and discontinuous Galerkin (dG(k)) time discretizations are applied to the nonstation-ary incompressible Navier-Stokes equations. In particular for the cGP(2) method with quadratic ansatz functions in time, which lead to 3rd order accuracy in the L2-norm and even to 4th order superconvergence in the endpoints of the time intervals, together with the finite element pair Q2/P disc 1 for the spatial approximation of velocity and pressure leading to a globally 3rd order scheme, we explain the algorithmic details as well as implementation aspects. All presented solvers are analyzed with respect to their numerical costs for two prototypical flow configurations