Finite element error estimation for parabolic optimal control with measurement data

Abstract

A prior error estimate is considered for the finite element (FE) approximation of a parabolic optimal control (POC) with spatial measurement data. We use conforming linear finite element to discretize the space for the state, piecewise constant for the control, and Euler method to discretize the time. The convergence order $O(h^{2-\frac{s}{2}}+k^{\frac{1}{2}})$ in the $L^2(0,T,L^2\left(\Omega \right))$-norm of state variable, co-state, and control variable are obtained. To validate our theory, numerical tests are executed.