A stabilized finite element method for the convection dominated diffusion optimal control problem

Abstract

In this paper, a stabilized finite element method for optimal control problems governed by a convection dominated diffusion equation is investigated. The state and the adjoint variables are approximated by piecewise linear continuous functions with bubble functions. The control variable either is approximated by piecewise linear functions (called the standard method) or is not discretized directly (called the variational discretization method). The stabilization term only depends on bubble functions, and the projection operator can be replaced by the difference of two local Gauss integrations. A priori error estimates for both methods are given and numerical examples are presented to illustrate the theoretical results.

Keywords:

• Convection dominated diffusion equation
• stabilized finite element method
• optimal control problem
• variational discretization

AMS Subject Classifications:

• 65M60
• 76D07
• 65M12

Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions which helped us to improve the manuscript.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of Z F Weng is partially supported by the Scientific Research Foundation of Huaqiao University [grant number 15BS307]. The work of J Z Yang is partially supported by National Natural Science Foundation of China [grant number 11171305], [grant number 91230203], and the research of X. Lu is partially supported by National Natural Science Foundation of China [grant number 91230108], [grant number 11471253].