Error analysis for finite element approximation of parabolic Neumann boundary control problems

Our aim is to study a posteriori error estimates for the finite element method of the parabolic boundary control problems on a bounded convex polygonal domain. For discretization, piecewise linear and continuous finite elements are used to approximate the state and the adjoint-state variables, while piecewise constant functions are employed to approximate the control variable. The backward Euler implicit scheme is applied to discretize the time derivative. An adaptation of a novel elliptic reconstruction technique plays a key role in deriving the error estimates. The residual type a posteriori error estimates for the state and adjoint-state variables are derived in the L-infinity(0,T; H-1(Omega))- norm. Furthermore, an error bound for the control variable is established in the L-infinity(0,T; L-2(Gamma))- norm. The numerical experiments illustrate the performance of the derived estimators. The adaptive mesh generated via the error indicators is very less in comparison to the uniform mesh, which shows the effectiveness of our derived estimators.