@article{Xu2023, 
author = {Changling Xu and Huilai Li},
title = {Two-grid methods of finite element approximation for parabolic integro-differential optimal control problems},
year = {2023},
journal = {Electronic Research Archive},
volume = {31},
number = {8},
pages = {4818-4842},
keywords = {parabolic integro-differential equations, finite element methods, a priori error estimates, two-grid, superconvergence},
url = {https://www.sciopen.com/article/10.3934/era.2023247},
doi = {10.3934/era.2023247},
abstract = {In this paper, we present a two-grid scheme of fully discrete finite element approximation for optimal control problems governed by parabolic integro-differential equations. The state and co-state variables are approximated by a piecewise linear function and the control variable is discretized by a piecewise constant function. First, we derive the optimal a priori error estimates for all variables. Second, we prove the global superconvergence by using the recovery techniques. Third, we construct a two-grid algorithm and discuss its convergence. In the proposed two-grid scheme, the solution of the parabolic optimal control problem on a fine grid is reduced to the solution of the parabolic optimal control problem on a much coarser grid; additionally, the solution of a linear algebraic system on the fine grid and the resulting solution maintain an asymptotically optimal accuracy. Finally, we present a numerical example to verify the theoretical results.}
}