## (A) Posteriori Error Estimates for Finite Element Discretizations of Parabolic Optimal Control Problems

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2022

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##### Abstract

The main objective of this thesis is to derive a posteriori error estimates for nite element
discretizations of optimal control problems governed by parabolic partial di erential equations. Both distributed and boundary parabolic optimal control problems are considered and analyzed.
We first study L1(0; T;L2())a posteriori error analysis for parabolic optimal control
problem (POCP) with distributed control. To discretize the state and co-state variables we use the piecewise linear and continuous nite elements, while the piecewise constant functions are used to discretize the control variable. The backward Euler scheme is applied for the time discretization. An elliptic reconstruction technique in conjunction with energy argument is used to derive a posteriori error estimates for the state and costate variables in the L1(0; T;L2( ))- norm. The rst-order necessary optimality condition is used to derive the error estimate for the control variable in the L1(0; T;L2( ))-norm. The second problem considers the POCP with distributed control and discusses a posteriori error analysis for both semi-discrete and fully discrete nite element method. The variational discretization is used to approximate the state and co-state variables with the piecewise linear and continuous functions, while the control variable is computed by using the implicit relation between the control and co-state variables. The temporal discretization is based on the backward Euler method. The key feature of this approach is not to discretize the control variable but to implicitly utilize the optimality conditions for the discretization of the control variable. We use the elliptic reconstruction technique in conjunction with heat kernel estimates for linear parabolic problem to derive a posteriori error estimates for the state, co-state and control variables in the L1(0; T; L1( ))-norm. Use of elliptic reconstruction technique greatly simpli es the analysis by allowing us to take the advantage of existing elliptic maximum norm error estimates and the heat kernel estimate. Our next problem focuses on nite element approximations of the POCP with controls acting on a lower dimensional manifold. The manifold considered here is either a point, or a curve or a surface which is lying completely in the space domain. In addition, the manifold is assumed to be either time independent or evolved with the time. The state and co-state variables are approximated by the piecewise linear and continuous nite elements whereas the
piecewise constant functions are employed to approximate the control variable. Moreover, the discrete-in-time scheme is based on the backward Euler method. We derive a posteriori error estimates for the various dimensions of the manifold.
We next turn our attention to study local a posteriori error estimates for the space-time
nite element approximations of parabolic boundary control problem (PBCP). In many engineering applications, it is often useful to study the behavior of the state and co-state variables in a small neighborhood of the boundary. Therefore, a posteriori error estimators in some suitable local norms have become more useful, and the derivation of these estimates is not straightforward. Therefore, an attempt has been made in this thesis to derive local a pos- teriori error estimates for the PBCP. The space-time discretization is accomplished by using the piecewise linear and continuous nite element approximations for the state and co-state variables while the piecewise constant function spaces employed for the control variable. We use the backward Euler method to approximate the time derivative. We derive three di erent reliable local a posteriori error estimates for Neumann boundary control problems with the observations of the boundary state, the distributed state and the nal state. Our derived estimators are of local character in the sense that the leading terms of the estimators depend on the small neighborhood of the boundary. These new local a posteriori error bounds can be used to study the behaviour of the state and co-state variables around the boundary and provide the necessary feedback in terms of the error indicators for the adaptive mesh re nements in the nite element method. Our last problem is devoted to study nite element approximation for nonlinear PBCP. The error analysis is carried out by using the piecewise linear and continuous nite elements for the approximation of the state and co-state variables whereas the approximation of the control variable is done with the piecewise constant functions. The backward Euler method is used to approximate the time derivative. The reliable type local a posteriori error bounds for the state, co-state and control variables in the L2(0; T;L2())- norm are derived, while the a posteriori error estimates for the control variable is established by assuming the second-order optimality condition.
Computational results are presented to illustrate the performance of the derived estimators.

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Supervisor: Rajen Kumar Sinha

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MATHEMATICS