A Priori Error Estimate of Splitting Positive Definite Mixed Finite Element Method for Parabolic Optimal Control Problems. Issue 2 (24th May 2016)

• Record Type: Journal Article
• Authors: Fu, Hongfei; Rui, Hongxing; Zhang, Jiansong; Guo, Hui
• Is Part Of: Numerical mathematics. Volume 9:Issue 2(2016)
• Page Start: 215
• Page End: 238
• Publication Date: 2016-05-24
• Subjects: 49K20, -- 65M60; Parabolic optimal control, -- splitting mixed finite element method, -- positive definite, -- a priori error estimates, -- numerical experiments; Numerical analysis -- Periodicals; Numerical analysis; Periodicals; 518.05
• DOI: 10.4208/nmtma.2016.m1409
• Languages: English
• ISSNs: 1004-8979

Abstract

Abstract: In this paper, we propose a splitting positive definite mixed finite element method for the approximation of convex optimal control problems governed by linear parabolic equations, where the primal state variable y and its flux σ are approximated simultaneously. By using the first order necessary and sufficient optimality conditions for the optimization problem, we derive another pair of adjoint state variables z and ω, and also a variational inequality for the control variable u is derived. As we can see the two resulting systems for the unknown state variable y and its flux σ are splitting, and both symmetric and positive definite. Besides, the corresponding adjoint states z and ω are also decoupled, and they both lead to symmetric and positive definite linear systems. We give some a priori error estimates for the discretization of the states, adjoint states and control, where Ladyzhenkaya-Babuska-Brezzi consistency condition is not necessary for the approximation of the state variable y and its flux σ. Finally, numerical experiments are given to show the efficiency and reliability of the splitting positive definite mixed finite element method.