Error Analysis for Parabolic Optimal Control Problems with Measure Data in a Nonconvex Polygonal Domain (Journal of Computational Mathematics, 2024)
This paper considers the finite element approximation to parabolic optimal controlproblems with measure data in a nonconvex polygonal domain. Such problems usuallypossess low regularity in the state variable due to the presence of measure data and thenonconvex nature of the domain. The low regularity of the solution allows the finiteelement approximations to converge at lower orders. We prove the existence, uniquenessand regularity results for the solution to the control problem satisfying the first orderoptimality condition. For our error analysis we have used piecewise linear elements for theapproximation of the state and co-state variables, whereas piecewise constant functionsare employed to approximate the control variable. The temporal discretization is based onthe implicit Euler scheme. We derive both a priori and a posteriori error bounds for thestate, control and co-state variables. Numerical experiments are performed to validate thetheoretical rates of convergence.