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

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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.
    Mathematics Subject Classification: Primary:65M60, 35B37; Secondary: 49J20.


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