L∞-error estimates of rectangular mixed finite element methods for bilinear optimal control problem

Abstract

In this paper, we investigate L∞-error estimates of the bilinear elliptic optimal control problem by rectangular Raviart–Thomas mixed finite element methods. The control variable enters the state equation as a coefficient. The state and the co-state variables are approximated by the Raviart–Thomas mixed finite elements of order k=1, and the control variable is approximated by piecewise linear functions. The L∞-error estimates are obtained for the control variable and coupled state variable, and the convergence rates of orders O(h2) and O(h32|lnh|12) are also gained for the control and state variables and the flux of the state and co-state variables, respectively. In addition, the performance of the error estimates is assessed by two numerical examples.