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.
KeywordsDiscontinuous Galerkin Method Posteriori Error Estimation Adjoint Problem Temporal Discretization Moderate Reynolds Number
Unable to display preview. Download preview PDF.
- 5.C. Brezinski, U. Ieea, J. V. Iseghem, A Taste of Pade Approximation, Acta Numerica 4:53-103, Cambridge University Press (1995)Google Scholar
- 10.T. Richter, A. Springer, B. Vexler, Efficient numerical realization of discontinuous Galerkin methods for temporal discretization of parabolic problems, accepted for Numerische Mathematik, 2012Google Scholar
- 11.M. Schäfer, S. Turek, Benchmark computations of laminar flow around a cylinder. (With support by F. Durst, E. Krause and R. Rannacher), Flow Simulation with High-Performance Computers II. DFG priority research program results 1993-1995, pp 547-566, Vieweg, Wiesbaden (1996)Google Scholar
- 12.M. Schmich, Adaptive Finite Element Methods for Nonstationary Incompressible Flow Problems, Dissertation, Universität Heidelberg (2009)Google Scholar
- 13.M. Schmich, B. Vexler, Adaptivity with dynamic meshes for space-time finite element discretizations of parabolic equations, SIAM J. Sci. Comput., Vol. 30, No. 1, pp. 369-393.Google Scholar
- 15.S. Turek, Efficient Solvers for Incompressible Flow Problems, Lecture Notes in Computational Science and Engineering 6, Springer, (1999)Google Scholar
- 16.The Finite Element Toolkit Gascoigne, http://www.gascoigne.uni-hd.de.