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

Abstract

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 ow: discretization, optimization and stability analysis". In the first lecture on "Numerical computation of incompressible viscous ow", we discuss the Galerkin finite element method for the discretization of the Navier-Stokes equations for modeling laminar ow. 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 ow control", we discuss the use of the DWR method for adaptive discretization in ow control and model calibration. Finally, in the fourth lecture on "Numerical stability analysis", we consider the numerical stability analysis of stationary ows employing the concepts of linearized stability and pseudospectrum.