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)cGP(k)) and discontinuous Galerkin (dG(k  )) time discretizations are applied to the nonstationary incompressible Navier–Stokes equations. In particular for the cGP(2)cGP(2) method with quadratic ansatz functions in time, which lead to 3rd order accuracy in the L2L2-norm and even to 4th order superconvergence in the endpoints of the time intervals, together with the finite element pair Q2/P1disc 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.