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

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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

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

Received May 2005 Revised March 2006 Published June 2006

Abstract
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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 & Continuous Dynamical Systems - B, 2006, 6 (5) : 1077-1096. doi: 10.3934/dcdsb.2006.6.1077

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