In this paper a posteriori error analysis of virtual element method (VEM) for the optimal control problem governed by general elliptic equation is presented. The virtual element discrete scheme is constructed with virtual element approximation of the state equation and variational discretization of the control variable. Based on the a posteriori error estimates of virtual element method for general elliptic equation and approximated error equivalence of the solution of the optimal control problem to solutions of the state and adjoint problems we build up upper and lower a posteriori error estimates of the optimal control problem. Under the Dörfler’s marking strategy, the traditional projected gradient algorithm and adaptive VEM algorithm drived by the state and adjoint error estimators are used to solve the optimal control problem. Numerical experiments are carried out to illustrate the theoretical findings.

本文针对一般椭圆方程控制的最优控制问题，提出了虚元法(VEM)的后验误差分析。虚元离散方案由状态方程的虚元逼近和控制变量的变分离散构成。基于一般椭圆方程虚元法的后验误差估计和最优控制问题解与状态和伴随问题解的近似误差等价，我们建立了最优控制问题的上下后验误差估计. 在 Dörfler 的标记策略下，采用传统的投影梯度算法和由状态和伴随误差估计器驱动的自适应 VEM 算法来解决最优控制问题。