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.

No abstract

Microwave heating has been widely used in various fields during recent years. However, it also has a common problem of uneven heating. In this paper, optimal frequency control problem for microwave heating process is considered. The cost function is defined such that the temperature profile at the final stage has a relative uniform distribution in the field. The controlled system is a coupled by Maxwell equations with nonlinear heating equation. The existence of a weak solution for coupled system is proved. The weak continuity of the solution operator is also shown. Moreover, the existence of a global minimizer of the optimal frequency control problems is proved.