TY - JOUR
T1 - A Priori Error Estimate of Splitting Positive Definite Mixed Finite Element Method for Parabolic Optimal Control Problems
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 - <p style="text-align: justify;">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.</p>