Superconvergence of splitting positive definite mixed finite element for parabolic optimal control problems

ABSTRACT

In this paper, we investigate the superconvergence of fully discrete splitting positive definite mixed finite element (MFE) methods for parabolic optimal control problems. For the space discretization, the state and co-state are approximated by the lowest order Raviart–Thomas MFE spaces and the control variable is approximated by piecewise constant functions. The time discretization of the state and co-state are based on finite difference methods. We derive the superconvergence between the projections of exact solutions and numerical solutions or the exact solutions and postprocessing numerical solutions for the control, state and co-state. A numerical example is provided to validate the theoretical results.

Keywords:

• Superconvergence
• parabolic equations
• optimal control problems
• splitting positive definite
• mixed finite element methods

AMS Subject Classifications:

• 49J20
• 65N30

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China [grant number 11401201]; the Hunan Province Education Department [grant number 16B105].