In this paper, we study the finite volume element method of bilinear parabolic optimal control problem. We will use the optimize-then-discretize approach to obtain the semi-discrete finite volume element scheme for the optimal control problem. Under some reasonable assumptions, we derive the optimal order error estimates in $L^2(J;L^2)$ and $L^{\mathrm{\infty }}(J;L^2)$-norm. We use the backward Euler method for the discretization of time to get fully discrete finite volume element scheme for the optimal control problem, and obtain some error estimates. The approximate order for the state, costate and control variables is $O(h^{3/2}+\mathrm{△}t)$ in the sense of $L^2(J;L^2)$ and $L^{\mathrm{\infty }}(J;L^2)$-norm. Finally, a numerical experiment is presented to test these theoretical results.

To address pollution control, cooperation among interested parties is essential. This paper examines a stochastic evolution game model involving a strategic coalition of three enterprises for pollution control. The model incorporates decision-making factors to construct a strategic coalition model of enterprise pollution control between enterprises. The study analyzes the evolution process of pollution control strategies within the coalition and determines the final stable strategy. Additionally, numerical simulations are conducted to explore the evolution paths of enterprises under various circumstances. Results from a case-based numerical example validate the theoretical findings, supporting the following insights: (1) the coalition requires an effective mechanism for incentivizing and penalizing actions, (2) the government should establish and improve coordination and management mechanisms, (3) citizens are urged to develop social oversight mechanisms for reporting environmental violation, and (4) a fair and equitable distribution mechanism should be implemented within the coalition.