@Article{NMTMA-9-215,
author = {},
title = {A Priori Error Estimate of Splitting Positive Definite Mixed Finite Element Method for Parabolic Optimal Control Problems},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2016},
volume = {9},
number = {2},
pages = {215--238},
abstract = {<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>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.2016.m1409},
url = {http://global-sci.org/intro/article_detail/nmtma/12375.html}
}