Interpolation coefficients mixed finite element methods for general semilinear Dirichlet boundary elliptic optimal control problems

ABSTRACT

In this paper, we study a priori error estimates of interpolation coefficients mixed finite element methods for semilinear Dirichlet boundary optimal control problems. Using the interpolation coefficient thought to process the nonlinear term of equations, we present the mixed finite element approximation with interpolated coefficients for general optimal control problems governed by semilinear Dirichlet boundary elliptic equations. The state variable and the co-state variable are discretized by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is discretized by piecewise constant elements. We derive a priori error estimates in norm and norm for the coupled state and control variables with optimal convergence order. Finally, some numerical examples are given to confirm our theoretical results.

Keywords:

• A priori error estimates
• semilinear Dirichlet boundary optimal control problems
• interpolation coefficients
• mixed finite element methods

AMS Subject Classifications:

• 49J20
• 65N30

Acknowledgements

The authors express their thanks to the referees for their helpful suggestions, which led to improvements of the presentation.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by Major Research Plan of National Natural Science Foundation of China [grant number 91430108], National Science Foundation of China [grant number 11201510], [grant number 11171251], Innovation Team Building at Institutions of Higher Education in Chongqing [grant number CXTDX201601035], China Postdoctoral Science Foundation [grant number 2017T100155], [grant number 2015M580197], Chongqing Research Program of Basic Research and Frontier Technology [grant number cstc2015jcyjA20001].