Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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)
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)
Brenner, S. C., \(C^0\) Interior Penalty Methods, Frontiers in Numerical Analysis-Durham 2010, Springer-Verlag, Berlin-Heidelberg, 85, 79–147, (2012)
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)
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)
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)
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)
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)
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)
Dauge, M., Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Mathematics 1341, Springer-Verlag, Berlin-Heidelberg, (1988)
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)
Frehse, J., Zum Differenzierbarkeitsproblem bei Variationsungleichungen höherer Ordnung. Abh. Math. Sem. Univ. Hamburg, 36, 140–149, (1971)
Friedman, A., Variational Principles and Free-Boundary Problems. Robert E. Krieger Publishing Co., Inc., Malabar, FL, second edition, (1988)
Gong, W., Yan, N., A mixed finite element scheme for optimal control problems with pointwise state constraints. J. Sci. Comput., 46, 182–203, (2011)
Grisvard, P., Elliptic Problems in Non Smooth Domains. Pitman, Boston (1985)
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)
A. Heroux, M. A., Willenbring, J. M., Trilinos Users Guide, Sandia National Laboratories, (2003)
Hintermüller, M., Ito, K., Kunisch, K., The primal-dual active set strategy as a semismooth Newton method., SIAM J. Optim., 13, 865–888, (2003)
Hinze, M., Pinnau, R.,Ulbrich, M., Ulbrich, S., Optimization with PDE Constraints, Springer, New York, (2009)
Ji, X., Sun, J., Yang, Y., Optimal penalty parameter for \(C^0\) IPDG. Appl. Math. Lett., 37, 112–117, (2014)
Kärkkäinen, T., Kunisch, K., Tarvainen, P., Augmented Lagrangian active set methods for obstacle problems. J. Optim. Theory Appl., 119, 499–533 (2003)
Kinderlehrer, D., Stampacchia, G., An Introduction to Variational Inequalities and Their Applications. Society for Industrial and Applied Mathematics, Philadelphia, (2000)
Lions, J.-L., Stampacchia, G., Variational inequalities. Comm. Pure Appl. Math., 20, 493–519, (1967)
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)
Maz’ya, V., Rossmann, J., Elliptic Equations in Polyhedral Domains. American Mathematical Society, Providence, RI, (2010)
Rodrigues, J.-F., Obstacle Problems in Mathematical Physics. North-Holland Publishing Co., Amsterdam, 134, (1987)
Acknowledgments
The work of the first author was supported in part by the National Science Foundation under Grant No. DMS-13-19172.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer Science+Business Media New York
About this chapter
Cite this chapter
Brenner, S.C., Oh, M., Pollock, S., Porwal, K., Schedensack, M., Sharma, N.S. (2016). A \(\varvec{C}^0\) Interior Penalty Method for Elliptic Distributed Optimal Control Problems in Three Dimensions with Pointwise State Constraints. In: Brenner, S. (eds) Topics in Numerical Partial Differential Equations and Scientific Computing. The IMA Volumes in Mathematics and its Applications, vol 160. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6399-7_1
Download citation
DOI: https://doi.org/10.1007/978-1-4939-6399-7_1
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-6398-0
Online ISBN: 978-1-4939-6399-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)