Higher order Galerkin–collocation time discretization with Nitsche’s method for the Navier–Stokes equations

We propose and study numerically the implicit approximation in time of the Navier–Stokes equations by a Galerkin–collocation method in time combined with inf–sup stable finite element methods in space. The conceptual basis of the Galerkin–collocation approach is the establishment of a direct connection between the Galerkin method and the classical collocation methods, with the perspective of achieving the accuracy of the former with reduced computational costs in terms of less complex algebraic systems of the latter. Regularity of higher order in time of the discrete solution is ensured further. As an additional ingredient, we employ Nitsche’s method to impose all boundary conditions in weak form with the perspective that evolving domains become feasible in the future. We carefully compare the performance properties of the Galerkin–collocation approach with a standard continuous Galerkin–Petrov method using piecewise linear polynomials in time, that is algebraically equivalent to the popular Crank–Nicholson scheme. The condition number of the arising linear systems after Newton linearization as well as the reliable approximation of the drag and lift coefficient for laminar flow around a cylinder (DFG flow benchmark with Re=100; cf. (Turek and Schäfer, 1996)) are investigated. The superiority of the Galerkin–collocation approach over the linear in time, continuous Galerkin–Petrov method is demonstrated therein.