Optimal a priori error estimates of parabolic optimal control problems with pointwise control

In this paper we consider a parabolic optimal control problem with a pointwise (Dirac type) control in space, but variable in time two space dimensions. To approximate the problem we use the standard continuous piecewise linear approximation in space and the piecewise constant discontinuous Galerkin method in time. Despite low regularity of the state equation, we show almost optimal h² + k convergence rate for the control in L² norm. This result improves almost twice the previously known estimate in [W. Gong, M. Hinze, and Z. Zhou, A Priori Error Analysis for Finite Element Approximation of Parabolic Optimal Control Problems with Pointwise Control, Tech. report, 2011-07, Hamburger Beitr̈age zur Angewandten Mathematik, Hamburg, Germany, 2011].