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Linear-quadratic optimal control for the Oseen equations withstabilized finite elements

Published online by Cambridge University Press:  16 January 2012

Malte Braack  and
Benjamin Tews
Malte Braack
Affiliation:
Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098 Kiel, Germany. [email protected]; [email protected]
Benjamin Tews
Affiliation:
Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098 Kiel, Germany. [email protected]; [email protected]
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Abstract

For robust discretizations of the Navier-Stokes equations with small viscosity, standardGalerkin schemes have to be augmented by stabilization terms due to the indefiniteconvective terms and due to a possible lost of a discrete inf-sup condition. For optimalcontrol problems for fluids such stabilization have in general an undesired effect in thesense that optimization and discretization do not commute. This is the case for thecombination of streamline upwind Petrov-Galerkin (SUPG) and pressure stabilizedPetrov-Galerkin (PSPG). In this work we study the effect of different stabilized finiteelement methods to distributed control problems governed by singular perturbed Oseenequations. In particular, we address the question whether a possible commutation error inoptimal control problems lead to a decline of convergence order. Therefore, we givea priori estimates for SUPG/PSPG. In a numerical study for a flow withboundary layers, we illustrate to which extend the commutation error affects theaccuracy.

Keywords

OseenNavier-Stokesoptimal controlfinite elementsstabilized methods
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 4 , October 2012 , pp. 987 - 1004
Copyright
© EDP Sciences, SMAI, 2012

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References

Abraham, F., Behr, M. and Heinkenschloss, M., The effect of stabilization in finite element methods for the optimal boundary control of the Oseen equations. Finite Elem. Anal. Des. 41 (2004) 229–251. Google Scholar
R. Becker and M. Braack, A two-level stabilization scheme for the Navier-Stokes equations, in Numerical Mathematics and Advanced Applications, ENUMATH 2003. edited by, M. Feistauer et al., Springer (2004) 123–130.
Becker, R. and Vexler, B., Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math. 106 (2007) 349–367. Google Scholar
Braack, M., Optimal control in fluid mechanics by finite elements with symmetric stabilization. SIAM J. Control Optim. 48 (2009) 672–687. Google Scholar
Braack, M. and Burman, E., Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method. SIAM J. Numer. Anal. 43 (2006) 2544–2566. Google Scholar
Braack, M., Burman, E., John, V. and Lube, G., Stabilized finite element methods for the generalized Oseen problem. Comput. Methods Appl. Mech. Engrg. 196 (2007) 853–866. Google Scholar
Brooks, A.N. and Hughes, T.J.R., Streamline upwind Petrov-Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 32 (1982) 199–259. Google Scholar
S.S. Collis and M. Heinkenschloss, Analysis of the streamline upwind/Petrov Galerkin method applied to the solution of optimal control problems. Technical report 02-01, Rice University, Houston, TX (2002).
Dedé, L. and Quarteroni, A., Optimal control and numercal adaptivity for advection-diffusion equations. ESIAM : M2AN 39 (2005) 1019–1040. Google Scholar
V. Girault and P.-A. Raviart, Finite Elements for the Navier Stokes Equations. Springer, Berlin (1986).
Heinkenschloss, M. and Leykekhman, D., Local error estimates for SUPG solutions of advection-dominated elliptic linear-quadratic optimal control problems. SIAM J. Numer. Anal. 47 (2010) 4607–4638. Google Scholar
Hinze, M., Yan, N. and Zhou, Z., Variational discretization for optimal control governed by convection dominated diffusion equations. J. Comput. Math. 27 (2009) 237–253. Google Scholar
Johnson, C. and Saranen, J., Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations. Math. Comput. 47 (1986) 1–18. Google Scholar
Lube, G. and Rapin, G., Residual-based stabilized higher-order FEM for a generalized Oseen problem. Math. Models Methods Appl. Sci. 16 (2006) 949–966. Google Scholar
Lube, G. and Rapin, G., Residual-based stabilized higher-order FEM for a generalized Oseen problem. Math. Models Methods Appl. Sci. 16 (2006) 949–966. Google Scholar
Lube, G. and Tews, B., Optimal control of singularly perturb advection-diffusion-reaction problems. Math. Models Appl. Sci. 20 (2010) 1–21. Google Scholar
Matthies, G., Skrzypacz, P. and Tobiska, L., A unified convergence analysis for local projection stabilisations applied ro the Oseen problem. ESAIM : M2AN 41 (2007) 713–742. Google Scholar
Yan, N. and Zhou, Z., A priori and a posteriori error estimates of streamline diffusion finite element method for optimal control problems governed by convection dominated diffusion equation. NMTMA 1 (2008) 297–320. Google Scholar
Yan, N. and Zhou, Z., A priori and a posteriori error analysis of edge stabilization Galerkin method for the optimal control problem governed by convection dominated diffusion equation. J. Comput. Appl. Math. 223 (2009) 198–217. Google Scholar