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

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 nonstationary 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 L 2 -norm and even to 4th order superconvergence in the endpoints of the time intervals, together with the finite element pair Q 2 / P 1 disc 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.