Discontinuous Galerkin as Time-Stepping Scheme for the Navier–Stokes Equations

In this work we describe a fast solution algorithm for the time dependent Navier–Stokes equations in the regime of moderate Reynolds numbers. Special to this approach is the underlying discretization: both for spatial and temporal discretization we apply higher order Galerkin methods. In space, standard Taylor-Hood like elements on quadrilateral or hexahedral meshes are used. For time discretization, we employ discontinuous Galerkin methods. This combination of Galerkin discretizations in space and time allows for a consistent variational space-time formulation of the Navier Stokes equations. This brings along the benefit of a well defined adjoint problem to be used for optimization methods based on the Euler-Lagrange approach and for adjoint error estimation methods. Special care is given to the solution of the algebraic systems. Higher order discontinuous Galerkin formulations in time ask for a coupled treatment of multiple solution states. By an approximative factorization of the system matrices we can reduce the complex system to a multi-step method employing only standard backward Euler like time steps.