Some error estimates of finite volume element method for parabolic optimal control problems

In this paper, the finite volume element method (FVEM) is applied to solve the distributed optimal control problems governed by parabolic equation. We use the method of variational discretization concept to approximate the problems. We consider a semi‐discrete and a fully discrete piecewise linear FVEMs. For the semi‐discrete method, the optimal order error estimates in continuous L ^∞ (J; L²) and L ^∞ (J; H¹)‐norm are obtained; the suboptimal error estimates in continuous L ^∞ (J; L ^∞ )‐norm are also obtained. For the fully discrete method, the optimal order error estimates in discrete L²(J; L²) and L ^∞ (J; L²)‐norm are derived. Numerical experiments are presented to test these theoretical results.