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September  2006, 6(5): 1077-1096. doi: 10.3934/dcdsb.2006.6.1077

Error estimates for time-discretizations for the velocity tracking problem for Navier-Stokes flows by penalty methods

Konstantinos Chrysafinos 1,

1. 

Institut für Numerische Simulation, Universität at Bonn, Wegelerstr 6, Bonn 53115, Germany

Received  May 2005 Revised  March 2006 Published  June 2006

Semi-discrete in time approximations of the velocity tracking problem are studied based on a pseudo-compressibility approach. Two different methods are used for the analysis of the corresponding optimality system. The first one, the classical penalty formulation, leads to estimates of order $k + \varepsilon$, under suitable regularity assumptions. The estimate is based on previously derived results for the solution of the unsteady Navier-Stokes problem by penalty methods (see e.g. Jie Shen [26]) and the Brezzi-Rappaz-Raviart theory (see e.g. [12]). The second one, based on the artificially compressible optimality system, leads to an improved estimate of the form $k + \varepsilon k$ for the linearized system.
Keywords: Navier Stokes equations, semidiscrete error estimates., artificial compressibility, velocity tracking, Optimal control.
Mathematics Subject Classification: Primary:65M60, 35B37; Secondary: 49J2.
Citation: Konstantinos Chrysafinos. Error estimates for time-discretizations for the velocity tracking problem for Navier-Stokes flows by penalty methods. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1077-1096. doi: 10.3934/dcdsb.2006.6.1077
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