| In this paper, we consider an optimal control problems for an Elliptic e-quation with L2control constraint, which is equivalent to the coupled system of state equation, co-state equation and the variational inequality, which is re-lated to the optimal control. Based on this, the adaptive finite element scheme is constructed and analyzed, then, the a priori error and a posteriori error are established. Finally, numerical examples are given to verify the effectiveness of the format.In chapter1. we introduce the background, significance of the research question, research situation and the content of this paper.In recent decades, the optimal control problem governed by partial differ-ential equation is a very important research direction. Adaptive finite element method has been widely used in many kinds of optimal control problem be-cause of its advantages, and based on this, both the ellipse and parabolic type problem have achieved significant improvement. In this paper, we consider an optimal control problems for an Elliptic equation with L2control constraint.In chapter2, we introduce the problem and testify the existence of the optimal control, which is as follows: then the finite element approximation is constructed: At last, we get the explicit solutions of variational inequalities: u=X(-p) and uh=λh(-Rhph).In chapter3, we obtain the a priori and a posteriori error estimate analysis of the problem respectively. The a priori error estimate is:‖y-yh‖1+‖p-ph‖1+‖u-uh‖0≤ch. And the a posteriori error estimate is:‖u-uh‖0,Ω2+‖y-yh‖1,Ω2+‖p-ph‖1,Ω2≤c∑ηi2and∑ηi2≤c(‖u-uh‖0,Ω2+‖y-yh‖1,Ω2+‖p-ph‖1,Ω2+ε2).In chapter4, we give detailed description of the gradient projection method, it is proved by the results of the numerical examples that the adaptive grid using a posteriori error estimation indicators to guide the mesh plays an im-port ant role in computational efficiency of the finite element.At last, we list the references involved in this paper. |
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