A Priori Error Estimate of Splitting Positive Definite Mixed Finite Element Method for Parabolic Optimal Control Problems
Year: 2016
Numerical Mathematics: Theory, Methods and Applications, Vol. 9 (2016), Iss. 2 : pp. 215–238
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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.2016.m1409
Numerical Mathematics: Theory, Methods and Applications, Vol. 9 (2016), Iss. 2 : pp. 215–238
Published online: 2016-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 24
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