Fourier Finite Volume Element Method for Two Classes of Optimal Control Problems Governed by Elliptic PDEs on Complex Connected Domain

Abstract

In this article, Fourier finite volume element method is provided to solve two classes of optimal control problems governed by elliptic PDEs. In two classes of problems, both distributed control and Dirichlet boundary control are given. We obtain the optimality systems of optimal control problems by the Lagrange multiplier method. Using the polar coordinates, the Fourier discretization is used in the azimuthal direction while the finite volume element approximation is used in the radial direction. In our proposed numerical method, the trial and test function spaces are carefully chosen to get accurate approximation. We use the Fourier finite volume method to approximate the control problems, the two-dimensional optimal control problem can be reduced to a one-dimensional problem. We not only observe that the convergent order of distributed control is second order, but also the convergent order of the Dirichlet boundary control is second order. Some numerical experiments are presented to illustrate accuracy and efficiency of our method.

Keywords:

• Fourier finite volume element method
• optimal control problems
• elliptic equation
• Dirichlet boundary control
• distributed control

Acknowledgments

We are very grateful to three anonymous referees for their valuable suggestions which have helped to improve the article.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China (Grant No 11971241), Hunan Provincial Innovation Foundation For Postgraduate (Grant No. CX2018B320).