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
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