Abstract
This article studies a finite element discretization of the optimal control problem governed by a parabolic equation in a convex polygonal domain. The control variable enters the state equation as a coefficient and is subject to the pointwise inequality constraints. We adopt optimize-then-discretize strategy to approximate the control problem. Both spatial and temporal discretizations of the state equations are considered and analyzed. The space discretization uses continuous piecewise linear finite elements for the approximation of the state variable and piecewise constant functions for the control variable. A linearized backward Euler scheme is used for the time discretization. We derive a priori error estimate in the L2(0,T;L2(Ω)) norm for the state and control variables for both the spatially discrete and fully-discrete schemes. Numerical experiment is performed to illustrate our theoretical results.
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