# 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 and Continuous Dynamical Systems - S, 2012, 5 (6) : 1147-1194. doi: 10.3934/dcdss.2012.5.1147
##### References:
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Optimization Control, 39 (2000), 113-132. doi: 10.1137/S0363012999351097. [11] R. Becker and R. Rannacher, Finite element solution of the incompressible Navier-Stokes equations on anisotropically refined meshes, Proc. Workshop Fast Solvers for Flow Problems, Kiel, 1994, Vieweg, 1995. [12] R. Becker and R. Rannacher, Weighted a posteriori error control in FE methods, Lecture at ENUMATH-95, Paris, Sept. 18-22, 1995, Proc. ENUMATH'97 (H. G. Bock et al., eds), pp. 621-637, World Scientific, Singapore, 1998. [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), 237-264. [14] R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer., 10 (2001), 1-102. doi: 10.1017/S0962492901000010. [15] R. Becker and B. 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##### References:
 [1] M. Ainsworth and J. T. Oden, A posteriori error estimation in finite element analysis, Comput. Methods Appl. Mech. Eng., 142 (1997), 1-88. doi: 10.1016/S0045-7825(96)01107-3. [2] W. Bangerth and R. Rannacher, "Adaptive Finite Element Methods for Differential Equations," Lectures in Mathematics, Birkhäuser, 2003. [3] R. Becker, An adaptive finite element method for the Stokes equations including control of the iteration error, in Numerical Mathematics and Advanced Applications, Proc. of ENUMATH 1997 (Eds. H.G. Bock et al.), pp. 609-620, World Scientific, London, 1998. [4] R. Becker, An optimal-control approach to a posteriori error estimation for finite element discretizations of the Navier-Stokes equations, East-West J. Numer. Math., 9 (2000), 257-274. [5] R. Becker, Mesh adaptation for stationary flow control, J. Math. Fluid Mech., 3 (2001), 317-341. doi: 10.1007/PL00000974. [6] R. Becker and M. Braack, A finite element pressure gradient stabilization for the Stokes equations based on local projection, Calcolo, 38 (2001), 363-379. doi: 10.1007/s10092-001-8180-4. [7] R. Becker, M. Braack, D. Meidner, R. Rannacher and B. Vexler, Adaptive finite element methods for PDE-constrained optimal control problems, in Reactive Flow, Diffusion and Transport (W. Jäger et al., eds), pp. 177-205, Springer, Heidelberg-Berlin, 2006. [8] R. Becker, M. Braack, R. Rannacher and Th. Richter, Mesh and model adaptivity for flow problems, in Reactive Flows, Diffusion and Transport (W. Jäger et al., eds), pp. 47-75, Springer, Heidelberg-Berlin, 2006. [9] R. Becker, V. Heuveline and R. Rannacher, An optimal control approach to adaptivity in computational fluid mechanics, Int. J. Numer. Meth. Fluids, 40 (2001), 105-120. doi: 10.1002/fld.269. [10] R. Becker, H. Kapp and R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: basic concepts, SIAM J. Optimization Control, 39 (2000), 113-132. doi: 10.1137/S0363012999351097. [11] R. Becker and R. Rannacher, Finite element solution of the incompressible Navier-Stokes equations on anisotropically refined meshes, Proc. Workshop Fast Solvers for Flow Problems, Kiel, 1994, Vieweg, 1995. [12] R. Becker and R. Rannacher, Weighted a posteriori error control in FE methods, Lecture at ENUMATH-95, Paris, Sept. 18-22, 1995, Proc. ENUMATH'97 (H. G. Bock et al., eds), pp. 621-637, World Scientific, Singapore, 1998. [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), 237-264. [14] R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer., 10 (2001), 1-102. doi: 10.1017/S0962492901000010. [15] R. Becker and B. 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