A splitting positive definite mixed finite element method for elliptic optimal control problem

摘要

In this paper, we propose a splitting positive definite mixed finite element method (MFEM) for the approximation of convex optimal control problem governed by elliptic equations with control constraints. By selecting the variation functional properly, the presented procedure can be split into two independent, symmetric and positive definite weak formula for the unknown state variable y and for the unknown flux variable σ. It then follows from the first order necessary and sufficient optimality condition, we deduce another two corresponding adjoint state equations for z and w, which are also independent, symmetric and positive definite. Also, a variational inequality for the control variable u is involved. Convergence analysis shows that the method yields the approximate solutions with optimal accuracy in L2(ΩU)-norm for the control u,L2(Ω)-norm for the original state y and adjoint state z,H(div;Ω)-norm for the flux state σ and adjoint state w without requiring the LBB consistency condition. Finally, some numerical examples are presented to confirm our theoretical results.