A \(\varvec{C}^0\) Interior Penalty Method for Elliptic Distributed Optimal Control Problems in Three Dimensions with Pointwise State Constraints

  • Susanne C. BrennerEmail author
  • Minah Oh
  • Sara Pollock
  • Kamana Porwal
  • Mira Schedensack
  • Natasha S. Sharma
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 160)


We investigate numerically a triquadratic \(C^0\) interior penalty method for elliptic distributed optimal control problems in three dimensions with pointwise state constraints, which is based on the formulation of these problems as fourth order variational inequalities. We obtain numerical results that are similar to the ones reported in [7, 8] for fourth order variational inequalities in two dimensions. The deal.II library [1, 2] is used for the numerical experiments.


Pointwise State Constraints Fourth Order Variational Inequalities Discrete Optimal State Post-processing Procedure Neumann Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The work of the first author was supported in part by the National Science Foundation under Grant No. DMS-13-19172.


  1. 1.
    Bangerth, W., Hartmann, R., Kanschat, G., deal.II – a General Purpose Object Oriented Finite Element Library. ACM Trans. Math. Softw., 33, 24/1–24/27, (2007)Google Scholar
  2. 2.
    Bangerth, W., Heister, T., Heltai, L., Kanschat, G., Kronbichler, M., Maier, M., Turcksin, B., Young, T. D., The deal.II Library, version 8.2. Archive of Numerical Software, 3, (2015)Google Scholar
  3. 3.
    Brenner, S. C., \(C^0\) Interior Penalty Methods, Frontiers in Numerical Analysis-Durham 2010, Springer-Verlag, Berlin-Heidelberg, 85, 79–147, (2012)Google Scholar
  4. 4.
    Brenner, S. C., Gu, S., Gudi, T., Sung, L.-Y., A quadratic \(C^0\) interior penalty method for linear fourth order boundary value problems with boundary conditions of the Cahn-Hilliard type. SIAM J. Numer. Anal., 50, 2088–2110, (2012)Google Scholar
  5. 5.
    Brenner, S. C., Neilan, M., A \(C^0\) interior penalty method for a fourth order elliptic singular perturbation problem. SIAM J. Numer. Anal., 49, 869–892, (2011)Google Scholar
  6. 6.
    Brenner, S. C., Sung, L.-Y., \(C^0\) interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput. 22/23, 83–118, (2005)Google Scholar
  7. 7.
    Brenner, S. C., Sung, L.-Y., Zhang, H., Zhang, Y., A quadratic \(C^0\) interior penalty method for the displacement obstacle problem of clamped Kirchhoff plates. SIAM J. Numer. Anal., 50, 3329–3350, (2012)Google Scholar
  8. 8.
    Brenner, S. C., Sung, L.-Y., Zhang, Y., A quadratic \(C^0\) interior penalty method for an elliptic optimal control problem with state constraints. Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations, Feng, X., Karakashian, O. and Xing, Y. eds.,The IMA Volumes in Mathematics and its Applications, Springer International Publishing, 157, 97–132, (2014)Google Scholar
  9. 9.
    Brenner, S. C., Sung, L.-Y., Zhang, Y., Post-processing procedures for an elliptic distributed optimal control problem with pointwise state constraints. Appl. Numer. Math., 95, 99–117, (2015)Google Scholar
  10. 10.
    Dauge, M., Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Mathematics 1341, Springer-Verlag, Berlin-Heidelberg, (1988)Google Scholar
  11. 11.
    Engel, G., Garikipati, K., Hughes, T. J. R., Larson, M. G., Mazzei, L., Taylor, R. L., Continuous/discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput. Methods Appl. Mech. Engrg., 191, 3669–3750, (2002)Google Scholar
  12. 12.
    Frehse, J., Zum Differenzierbarkeitsproblem bei Variationsungleichungen höherer Ordnung. Abh. Math. Sem. Univ. Hamburg, 36, 140–149, (1971)Google Scholar
  13. 13.
    Friedman, A., Variational Principles and Free-Boundary Problems. Robert E. Krieger Publishing Co., Inc., Malabar, FL, second edition, (1988)Google Scholar
  14. 14.
    Gong, W., Yan, N., A mixed finite element scheme for optimal control problems with pointwise state constraints. J. Sci. Comput., 46, 182–203, (2011)Google Scholar
  15. 15.
    Grisvard, P., Elliptic Problems in Non Smooth Domains. Pitman, Boston (1985)Google Scholar
  16. 16.
    Gudi, T., Gupta, H., Nataraj, N., Analysis of an interior penalty method for fourth order problems on polygonal domains. J. Sci. Comp. 54, 177–199 (2013)Google Scholar
  17. 17.
    A. Heroux, M. A., Willenbring, J. M., Trilinos Users Guide, Sandia National Laboratories, (2003)Google Scholar
  18. 18.
    Hintermüller, M., Ito, K., Kunisch, K., The primal-dual active set strategy as a semismooth Newton method., SIAM J. Optim., 13, 865–888, (2003)Google Scholar
  19. 19.
    Hinze, M., Pinnau, R.,Ulbrich, M., Ulbrich, S., Optimization with PDE Constraints, Springer, New York, (2009)Google Scholar
  20. 20.
    Ji, X., Sun, J., Yang, Y., Optimal penalty parameter for \(C^0\) IPDG. Appl. Math. Lett., 37, 112–117, (2014)Google Scholar
  21. 21.
    Kärkkäinen, T., Kunisch, K., Tarvainen, P., Augmented Lagrangian active set methods for obstacle problems. J. Optim. Theory Appl., 119, 499–533 (2003)Google Scholar
  22. 22.
    Kinderlehrer, D., Stampacchia, G., An Introduction to Variational Inequalities and Their Applications. Society for Industrial and Applied Mathematics, Philadelphia, (2000)Google Scholar
  23. 23.
    Lions, J.-L., Stampacchia, G., Variational inequalities. Comm. Pure Appl. Math., 20, 493–519, (1967)Google Scholar
  24. 24.
    Liu, W., Gong, W., Yan, N., A new finite element approximation of a state-constrained optimal control problem. J. Comput. Math., 27, 97–114, (2009)Google Scholar
  25. 25.
    Maz’ya, V., Rossmann, J., Elliptic Equations in Polyhedral Domains. American Mathematical Society, Providence, RI, (2010)Google Scholar
  26. 26.
    Rodrigues, J.-F., Obstacle Problems in Mathematical Physics. North-Holland Publishing Co., Amsterdam, 134, (1987)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Susanne C. Brenner
    • 1
    Email author
  • Minah Oh
    • 2
  • Sara Pollock
    • 3
  • Kamana Porwal
    • 1
  • Mira Schedensack
    • 4
  • Natasha S. Sharma
    • 5
  1. 1.Department of Mathematics and Center for Computation and TechnologyLouisiana State UniversityBaton RougeUSA
  2. 2.Department of Mathematics and StatisticsJames Madison UniversityHarrisonburgUSA
  3. 3.Department of MathematicsTexas A&M UniversityCollege StationUSA
  4. 4.Institut für Numerische SimulationUniversität BonnBonnGermany
  5. 5.Department of Mathematical SciencesUniversity of Texas of El PasoEl PasoUSA

Personalised recommendations