A FrameWork for the Error Analysis of Discontinuous Finite Element Methods for Elliptic Optimal Control Problems and Applications to C⁰ IP Methods

Abstract

In this article, an abstract framework for the error analysis of discontinuous Galerkin methods for control constrained optimal control problems is developed. The analysis establishes the best approximation result from a priori analysis point of view and delivers a reliable and efficient a posteriori error estimator. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posedness of the problem. Subsequently, the applications of C ⁰ interior penalty methods for a boundary control problem as well as a distributed control problem governed by the biharmonic equation subject to simply supported boundary conditions are discussed through the abstract analysis. Numerical experiments illustrate the theoretical findings.

Keywords:

• Biharmonic
• C ⁰IP method
• Discontinuous Galerkin
• Error bounds
• Finite element
• Optimal control
• Simply supported plate

Mathematics Subject Classification:

• 65N30
• 65N15

ACKNOWLEDGEMENTS

The authors would like to thank the referees for their comments and suggestions which have helped the authors to improve the presentation. The first author would like to acknowledge the support from NBHM-India, the second author would like to acknowledge the support from DST fast track project and all the authors would like to thank the UGC Center for Advanced Study.

Notes

Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/lnfa.