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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R. Becker, M. Braack, R. Rannacher and Th. Richter, Mesh and model adaptivity for flow problems,, in Reactive Flows, (2006), 47. Google Scholar

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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. doi: 10.1002/fld.269. Google Scholar

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[12]

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[13]

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[14]

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[15]

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[16]

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[17]

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[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

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[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]

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[26]

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[27]

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[28]

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[31]

Th. Dunne and R. Rannacher, Adaptive finite element approximation of fluid-structure interaction based on an Eulerian variational formulation,, in Fluid-Structure Interaction: Modelling, (2006), 110. Google Scholar

[32]

W. E and J. G. Liu, Projection method I: convergence and numerical boundary layers,, SIAM J. Numer. Anal., 32 (1995), 1017. doi: 10.1137/0732047. Google Scholar

[33]

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