Distributed Optimal Control of Diffusion-Convection-Reaction Equations Using Discontinuous Galerkin Methods

  • H. Yücel
  • M. Heinkenschloss
  • B. Karasözen
Conference paper


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.


Optimal Control Problem Discontinuous Galerkin Discontinuous Galerkin Method Adjoint Equation Interior Penalty 
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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.


  1. 1.
    Arnold D., Brezzi F., Cockborn B., Marini L.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002).MATHCrossRefGoogle Scholar
  2. 2.
    Ayuso B., Marini L. D.: Discontinuous Galerkin methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 47, 1391–1420 (2009).MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Becker R., Vexler B.: Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math. 106, 349–367 (2007).MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Castillo P.: Performance of Discontinuous Galerkin Methods for Elliptic PDEs. SIAM J. Sci. Comput. 24, 624–547 (2002).MathSciNetCrossRefGoogle Scholar
  5. 5.
    Collis S. S., Heinkenschloss M.: Analysis of the streamline upwind/Petrov Galerkin method applied to the solution of optimal control problems. Tech. Rep. TR02-01, Department of Computational and Applied Mathematics, Rice University, (2002).Google Scholar
  6. 6.
    Heinkenschloss M., Leykekhman D.: Local error estimates for SUPG solutions of advection-dominated elliptic linear-quadratic optimal control problems. SIAM J. Numer. Anal. 47, 4607–4638 (2010).MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Houston P., Schwab C., Süli E.: Discontinuous hp-Finite Element Methods for Advection-Diffusion-Reaction Problems. SIAM J. Numer. Anal. 39, pp. 2133–2163 (electronic) (2002).Google Scholar
  8. 8.
    Leykekhman D., Heinkenschloss M.: Local error analysis of discontinuous Galerkin methods for advection-dominated elliptic linear-quadratic optimal control problems. SIAM J. Numer. Anal. 50, 2012–2038 (2012).MATHCrossRefGoogle Scholar
  9. 9.
    Leykekhman D.: Investigation of Commutative Properties of Discontinuous Galerkin Methods in PDE Constrained Optimal Control Problems. J. of Scientific Computing, 1–29 (2012).Google Scholar
  10. 10.
    Rivière B.: Discontinuous Galerkin methods for solving elliptic and parabolic equations. Theory and implementation. SIAM Volume 35 of Frontiers in Applied Mathematics, (2008).Google Scholar
  11. 11.
    Yan, N, Zhou, Z.: A priori and a posteriori error analysis of edge stabilization Galerkin method for the optimal control problems governed by convection-dominated diffusion equation, Journal of Computational and Applied Mathematics 223, 198–217 (2009).MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Zhou Z., Yan N.: The local discontinuous Galerkin method for optimal control problem governed by convection-diffusion equations. International Journal of Numerical Analysis & Modeling 7, 681–699 (2010).MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Mathematics and Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey
  2. 2.Department of Computational and Applied MathematicsRice UniversityHoustonUSA

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