Abstract
We discuss the symmetric interior penalty Galerkin (SIPG) method, the nonsymmetric interior penalty Galerkin (NIPG) method, and the incomplete interior penalty Galerkin (IIPG) method for the discretization of optimal control problems governed by linear diffusion-convection-reaction equations. For the SIPG discretization the discretize-then-optimize (DO) and the optimize-then-discretize (OD) approach lead to the same discrete systems and in both approaches the observed L 2 convergence for states and controls is \(O({h}^{k+1})\), where k is the degree of polynomials used. The situation is different for NIPG and IIPG, where the the DO and the OD approach lead to different discrete systems. For example, when standard penalization is used, the L 2 error in the controls is only O(h) independent of k. However, if superpenalization is used, the lack of adjoint consistency is reduced and the observed convergence for NIPG and IIPG is essentially equal to that of the SIPG method in the DO and OD approach.
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Acknowledgements
HY has been supported by the 2214-International Doctoral Research Fellowship Program TÜBITAK during his studies in the Department of Computational and Applied Mathematics, Rice University, Houston. The work of MH was supported in part by NSF DMS-0915238. BK was supported through a Fulbright Scholarship.
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Yücel, H., Heinkenschloss, M., Karasözen, B. (2013). Distributed Optimal Control of Diffusion-Convection-Reaction Equations Using Discontinuous Galerkin Methods. In: Cangiani, A., Davidchack, R., Georgoulis, E., Gorban, A., Levesley, J., Tretyakov, M. (eds) Numerical Mathematics and Advanced Applications 2011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33134-3_42
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DOI: https://doi.org/10.1007/978-3-642-33134-3_42
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