@article{Xu2022, 
author = {Changling Xu and Hongbo Chen},
title = {A two-grid        P    0    2  -       P    1   mixed finite element scheme for semilinear elliptic optimal control problems},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {4},
pages = {6153-6172},
keywords = {semilinear elliptic equations, distributed optimal control problems, two-grid, superconvergence, P02-P1 mixed finite element},
url = {https://www.sciopen.com/article/10.3934/math.2022342},
doi = {10.3934/math.2022342},
abstract = {This paper aims to construct a two-grid mixed finite element scheme for distributed optimal control governed by semilinear elliptic equations. The state and co-state are approximated by the        P    0    2  -       P    1   pair and the control variable is approximated by the piecewise constant functions. First, a superclose result for the control variable and a priori error estimates for all variables are obtained. Second, a two-grid        P    0    2  -       P    1   mixed finite element algorithm is presented and the corresponding error is analyzed. In the two-grid scheme, the solution of the semilinear elliptic optimal control problem on a fine grid is reduced to the solution of the semilinear elliptic optimal control problem on a much coarser grid and the solution of a linear decoupled algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy. We find that the two-grid method achieves the same convergence property as the        P    0    2  -       P    1   mixed finite element method if the two mesh sizes satisfy    h  =      H    2  . Finally, a numerical example demonstrating our theoretical results is presented.}
}