TY - JOUR T1 - A Priori Error Estimate of Splitting Positive Definite Mixed Finite Element Method for Parabolic Optimal Control Problems AU - Hongfei Fu, Hongxing Rui, Jiansong Zhang & Hui Guo JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 215 EP - 238 PY - 2016 DA - 2016/09 SN - 9 DO - http://doi.org/10.4208/nmtma.2016.m1409 UR - https://global-sci.org/intro/article_detail/nmtma/12375.html KW - AB -
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.