Local a posteriori error estimates for boundary control problems governed by nonlinear parabolic equations

We derive local a posteriori error estimates of finite element approximation to nonlinear parabolic boundary control problems in a bounded convex domain with Lipschitz boundary. We apply piecewise linear and continuous finite elements to approximate the state and co-state variables, and the control variable is approximated by using the piecewise constant functions. The discrete-in-time scheme is based on the backward Euler method. We derive a reliable type local a posteriori error estimates for Neumann boundary control problems in the L-2([0, T]; L-2(partial derivative Omega))-norm. The main feature of our estimators is that they are of local character in the sense that the leading terms of the estimators depend on the small neighbourhood of the boundary. These new local a posteriori error bounds can be used to study the behaviour of the state and co-state variables near the boundary. The derived error indicators will provide the necessary feedback for the adaptive mesh refinements in the finite element method. We report numerical tests to illustrate the effectiveness of our error indicators.(c) 2022 Elsevier B.V. All rights reserved.