Abstract
In this paper, we propose a novel stabilized mixed finite element method for the approximation of optimal control problems governed by reaction-diffusion equations. Compared with the classical mixed finite element methods, the main contributions of this paper are as follows. First, the novel method uses only one stabilization parameter which is not mesh-dependent, and, the new mixed bilinear formulation is coercive and continuous. Second, the novel method is easy to be implemented on a computer using the standard Lagrange finite element. Third, the solutions of the novel method to the optimal control problems require low regularities. Fourth, the Ladyzhenkaya-Babuska-Brezzi (LBB) or the inf-sup condition for the mixed element spaces is unnecessary. Based on the novel method, we derive both continuous and discrete optimality systems for the corresponding constrained optimal control problems, and then a priori error analysis in a weighted norm is discussed. Finally, numerical experiments are given to confirm the efficiency and reliability of the novel stabilized method.
Highlights
Optimal control problems and their finite element solutions are attracting increasingly attentions of scientists and engineers
A priori error estimates of finite element approximations for optimal control problems governed by linear elliptic equations were studied extensively; see, for example, Refs. [, – ]
Mixed finite element method has been found very useful in solving such optimal control problems which contain the flux state variable, see Refs. [ – ]
Summary
Optimal control problems and their finite element solutions are attracting increasingly attentions of scientists and engineers. All these works need a matched mixed finite element spaces for the state variable and its flux, i.e., LBB stability condition must be strictly satisfied. The novel method is proved to be efficient and reliable both in theoretical error analysis and numerical tests It embodies some advantages compared with the former work [ ]: It uses only one stabilization parameter which is not mesh-dependent, the corresponding mixed bilinear formulation is still coercive and continuous; it is easier to be implemented using the standard Lagrange finite elements; it has relatively lower regularity requirements of the solutions to optimal control problems; and it can be extended to solve optimal control problems governed by other type partial differential equations.
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