# American Institute of Mathematical Sciences

December  2012, 5(6): 1147-1194. doi: 10.3934/dcdss.2012.5.1147

## A short course on numerical simulation of viscous flow: Discretization, optimization and stability analysis

 1 Institute of Applied Mathematics, University of Heidelberg, Im Neuenheimer Feld 293/294, D-69120 Heidelberg, Germany

Received  November 2011 Revised  March 2012 Published  August 2012

This article contains part of the material of four introductory lectures given at the 12th school Mathematical Theory in Fluid Mechanics'', Spring 2011, at Kácov, Czech Republic, on Numerical simulation of viscous flow: discretization, optimization and stability analysis''. In the first lecture on Numerical computation of incompressible viscous flow'', we discuss the Galerkin finite element method for the discretization of the Navier-Stokes equations for modeling laminar flow. Particular emphasis is put on the aspects pressure stabilization and truncation to bounded domains. In the second lecture on Goal-oriented adaptivity'', we introduce the concept underlying the Dual Weighted Residual (DWR) method for goal-oriented residual-based adaptivity in solving the Navier-Stokes equations. This approach is presented for stationary as well as nonstationary situations. In the third lecture on Optimal flow control'', we discuss the use of the DWR method for adaptive discretization in flow control and model calibration. Finally, in the fourth lecture on Numerical stability analysis'', we consider the numerical stability analysis of stationary flows employing the concepts of linearized stability and pseudospectrum.
Citation: Rolf Rannacher. A short course on numerical simulation of viscous flow: Discretization, optimization and stability analysis. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1147-1194. doi: 10.3934/dcdss.2012.5.1147
##### References:
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##### References:
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Rannacher, Finite element solution of the incompressible Navier-Stokes equations on anisotropically refined meshes,, Proc. Workshop Fast Solvers for Flow Problems, (1994). Google Scholar [12] R. Becker and R. Rannacher, Weighted a posteriori error control in FE methods,, Lecture at ENUMATH-95, (1995), 18. Google Scholar [13] R. Becker and R. Rannacher, A feed-back approach to error control in finite element methods: basic analysis and examples,, East-West J. Numer. Math., 4 (1996), 237. Google Scholar [14] R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods,, Acta Numer., 10 (2001), 1. doi: 10.1017/S0962492901000010. Google Scholar [15] R. Becker and B. Vexler, A posteriori error estimation for finite element discretization of parameter identification problems,, Numer. Math., 96 (2004), 435. doi: 10.1007/s00211-003-0482-9. Google Scholar [16] R. Becker, C. Johnson and R. Rannacher, Adaptive error control for multigrid finite element methods,, Computing, 55 (1995), 271. doi: 10.1007/BF02238483. Google Scholar [17] C. Bertsch and V. Heuveline, On multigrid methods for the eigenvalue computation of non-selfadjoint elliptic operators,, East-West J. Numer. Math., 8 (2000), 275. Google Scholar [18] M. Besier and R. Rannacher, Goal-oriented space-time adaptivity in the finite element Galerkin method for the computation of nonstationary incompressible flow,, Int. J. Numer. Meth. Fluids, (2011). Google Scholar [19] M. Besier and W. Wollner, On the pressure approximation in nonstationary incompressible flow simulations on dynamically varying spatial meshes,, Int. J. Numer. Methods Fluids, (2011). Google Scholar [20] S. Bönisch, Th. Dunne and R. Rannacher, Lecture on numerical simulation of liquid-structure interaction,, in Hemodynamical Flows: Aspects of Modeling, (2007). Google Scholar [21] M. Braack and A. Ern, A posteriori control of modeling errors and discretization errors,, Multiscale Model. Simul., 1 (2003), 221. doi: 10.1137/S1540345902410482. Google Scholar [22] M. Braack, E. Burman, V. John and G. Lube, Stabilized finite element methods for the generalized Oseen equations,, Comput. Methods Appl. Mech. Engrg., 196 (2007), 853. doi: 10.1016/j.cma.2006.07.011. Google Scholar [23] M. Braack and R. Rannacher, "Adaptive Finite Element Methods for Low-Mach-Number Flows with Chemical Reactions,", Lecture Series 1999-03, (1999), 1999. Google Scholar [24] M. Braack and Th. Richter, Solutions of 3D Navier-Stokes benchmark problems with adaptive finite elements,, Computers and Fluids, 35 (2006), 372. doi: 10.1016/j.compfluid.2005.02.001. Google Scholar [25] J. H. Bramble and J. E. Osborn, Rate of convergence estimates for nonselfadjoint eigenvalue approximations,, Math. Comp., 27 (1973), 525. doi: 10.1090/S0025-5718-1973-0366029-9. Google Scholar [26] S. C. Brenner and R. L. 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