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SUMMARYIn 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. Copyright © 2012 John Wiley & Sons, Ltd.

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Abstract. In this paper, we discuss the a posteriori error estimates of the semidiscrete mixed finite element methods for quadratic optimal control problems governed by linear hyperbolic equations. The state and the co-state are discretized by the order k Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order k(k ≥ 0). Using mixed elliptic reconstruction method, a posteriori L ∞ (L 2 )-error estimates for both the state and the control approximation are derived. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem.