TY - JOUR
T1 - A Splitting Least-Squares Mixed Finite Element Method for Elliptic Optimal Control Problems
AU - H.-F. Fu, H.-X. Rui, H. Guo, J.-S. Zhang & J. Hou
JO - International Journal of Numerical Analysis and Modeling
VL - 4
SP - 610
EP - 626
PY - 2016
DA - 2016/07
SN - 13
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/455.html
KW - Optimal control, splitting least-squares, mixed finite element method, positive definite, a priori error estimates.
AB - <p style="text-align: justify;">In this paper, we propose a splitting least-squares mixed finite element method for
the approximation of elliptic optimal control problem with the control constrained by pointwise
inequality. By selecting a properly least-squares minimization functional, we derive equivalent two
independent, symmetric and positive definite weak formulation for the primal state variable and
its flux. Then, using the first order necessary and also sufficient optimality condition, we deduce
another two corresponding adjoint state equations, which are both independent, symmetric and
positive definite. Also, a variational inequality for the control variable is involved. For the
discretization of the state and adjoint state equations, either RT mixed finite element or standard $C^0$ finite element can be used, which is not necessary subject to the Ladyzhenkaya-Babuska-Brezzi
condition. Optimal a priori error estimates in corresponding norms are derived for the control,
the states and adjoint states, respectively. Finally, we use some numerical examples to validate
the theoretical analysis.</p>