The Convergence and Superconvergence of a MFEM For Elliptic Optimal Control Problems

Authors

Hongbo Guan, Zhengzhoug University of Light IndustryFollow
Yong Yang, Nanjing University of Aeronautics and AstronauticsFollow
Huiqing Zhu, University of Southern MississippiFollow

Document Type

Article

Publication Date

4-1-2020

Department

Mathematics

School

Mathematics and Natural Sciences

Abstract

In this paper, we investigate a mixed finite element method (MFEM) for the elliptic optimal control problems (OCPs) with a distributive control. The state variable and adjoint state variable are approximated by the conforming rectangular Q₁₁ + Q₀₁ × Q₁₀ elements pair. The discrete B-B condition is satisfied automatically, which is usually considered to be the key point of the MFEM. The control is then obtained by the orthogonal projection through the adjoint state. Optimal orders of convergence are derived for the above mentioned variables. Furthermore, superclose and superconvergence results are also established under certain reasonable regularity assumptions. Some numerical results are provided to verity the theoretical analysis. At last, the proposed method is extended to some other low order conforming and nonconforming elements.

Publication Title

Advances in Applied Mathematics and Mechanics

Volume

12

Issue

2

First Page

527

Last Page

544

Recommended Citation

Guan, H., Yang, Y., Zhu, H. (2020). The Convergence and Superconvergence of a MFEM For Elliptic Optimal Control Problems. Advances in Applied Mathematics and Mechanics, 12(2), 527-544.
Available at: https://aquila.usm.edu/fac_pubs/17916