An efficient and stable finite element solver of higher order in space and time for nonstationary incompressible flow

In this paper, we extend our work for the heat equation [1] and for the Stokes equations [2] to the nonstationary Navier-Stokes equations. We present fully implicit continuous Galerkin-Petrov (cGP) and discontinuous Galerkin (dG) time stepping schemes for incompressible flow problems which are, in contrast to standard approaches like for instance the Crank-Nicolson scheme, of higher order in time. In particular, we implement and analyze numerically the higher order dG(1) and cGP(2)-methods which are super-convergent of 3rd, resp., 4th order in time, while for the space discretization, the well-known LBB-stable finite element pair Q2/P disc 1 is used. The discretized systems of nonlinear equations are treated by using the Newton method, and the associated linear subproblems are solved by means of a monolithic multigrid method with a blockwise Vanka-like smoother [3]. We perform nonstationary simulations for two benchmarking configurations to analyze the temporal accuracy and efficiency of the presented time discretization schemes. As a first test problem, we consider a classical flow around cylinder benchmark [4]. Here, we concentrate on the nonstationary behavior of the flow patterns with periodic oscillations and examine the ability of the different time discretization schemes to capture the dynamics of the flow. As a second test case, we consider the nonstationary flow through a Venturi pipe [5, 6]. The objective of this simulation is to control the instantaneous and mean flux through this device. Copyright c © 2011 John Wiley & Sons, Ltd.