Volume 9, Issue 2
A Priori Error Estimate of Splitting Positive Definite Mixed Finite Element Method for Parabolic Optimal Control Problems

Hongfei Fu ,  Hongxing Rui and Jiansong Zhang & Hui Guo

10.4208/nmtma.2016.m1409

Numer. Math. Theor. Meth. Appl., 9 (2016), pp. 215-238.

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  • 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 LadyzhenkayaBabuska-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.

  • History

Published online: 2016-09

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