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In this paper we analyze an optimization problem with limited observationgoverned by a convection--diffusion--reaction equation. Motivated by a Schurcomplement approach, we arrive at continuous norms that enable analysis ofwell-posedness and subsequent derivation of error analysis and a preconditionerthat is robust with respect to the parameters of the problem. We provideconditions for inf-sup stable discretizations and present one suchdiscretization for box domains with constant convection. We also provide apriori error estimates for this discretization. The preconditioner requires afourth order problem to be solved. For this reason, we use IsogeometricAnalysis as a method of discretization. To efficiently realize thepreconditioner, we consider geometric multigrid with a standard Gauss-Seidelsmoother as well as a new macro Gauss-Seidel smoother. The latter smootherprovides good results with respect to both the geometry mapping and thepolynomial degree.

Robust preconditioning and error estimates for optimal control of the  convection-diffusion-reaction equation with limited observation in  Isogeometric analysis