Global superconvergence analysis of a nonconforming FEM for Neumann boundary OCPs with elliptic equations

Abstract

In this paper, a nonconforming finite element method (FEM) is proposed for the Neumann type boundary optimal control problems (OCPs) governed by elliptic equations. The state and adjoint state are approximated by the nonconforming $E{Q}_1^{\mathrm{rot}}$ elements, and the control is approximated by the orthogonal projection through the adjoint state. Some superclose behaviors are derived by full use of the distinguish characters of this $E{Q}_1^{\mathrm{rot}}$ element, such as the interpolation operator equals to the Ritz projection, and the consistency error is higher than its interpolation error in the broken energy norm. After that, the global superconvergence results are obtained by employing the so-called post-interpolation technique. Finally, some numerical results are provided to verify the theoretical analysis.

Keywords:

• Nonconforming FEMs
• Neumann boundary OCPs
• error estimates
• supercloseness
• superconvergence

2010 Mathematics Subject Classifications:

• 65N30
• 65N15

Acknowledgements

The authors are grateful to the two anonymous referees for their helpful suggestions, which led to a clear presentation.

Disclosure statement

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 (No. 11501527), PhD Start-up Project of ZZULI (2015BSJJ070), Excellent Young Scholars Foundation of ZZULI (No. 2016XGGJS008).