Finite Element Analysis of Optimal Control Problems Governed by Certain PDEs

Abstract

The primary goal of this thesis is to study finite element based \textit{a priori} and \textit{a posteriori} error analysis of optimal control problems of various kinds governed by linear elliptic PDEs of second order. This thesis studies interior and boundary control (Neumann and Dirichlet) problems. This thesis contains six chapters including an introductory chapter and a concluding chapter. Some preliminary and fundamental results of finite element methods and optimal control problems which play key roles for the subsequent analysis are reviewed in the introductory chapter. This is followed by a brief literature survey of the finite element based numerical analysis of PDE constrained optimal control problems. We conclude the introductory chapter with a discussion on the outline of the thesis.

In chapter \ref{chapt:Stokes_Distributed_Control}, an abstract framework for the error analysis of discontinuous finite element method is developed for the distributed and Neumann boundary control problems governed by the stationary Stokes equation with control constraints. {\it A~priori} error estimates of optimal order are derived for velocity and pressure in the energy norm and the $L^2$-norm, respectively. Moreover, a reliable and efficient {\it a~posteriori} error estimator is derived. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posedness of the problem. In particular, we consider the abstract results with suitable stable pairs of velocity and pressure spaces like as the lowest-order Crouzeix-Raviart finite element and piecewise constant spaces, piecewise linear and constant finite element spaces.

We study an energy space-based approach for the Dirichlet boundary optimal control problem governed by the Poisson equation with control constraints in chapter \ref{chap:Dirichlet_Diffusion_control}. The optimality system results in a simplified Signorini type problem for control which is coupled with boundary value problems for state and costate variables. We propose a finite element based numerical method using the linear Lagrange finite element spaces with discrete control constraints at the Lagrange nodes. The analysis is presented in a combination for both the gradient and the $L^2$ cost functional. {\em A priori} error estimates of optimal order in the energy norm is derived up to the regularity of the solution for both the cases.

In chapter \ref{chapt:Stokes_Dirichlet_Control}, we propose and analyze an energy space based approach to formulate the Dirichlet boundary optimal control problem governed by the Stokes equation. Most of the previous work in the Stokes Dirichlet boundary control problem, deals with either tangential control or the case where flux of control is zero. This choice of control is very particular and their choice of the formulation leads the control with limited regularity. To overcome this difficulty we introduce the Stokes problem with mixed boundary conditions and the control acts on the Dirichlet boundary only hence our control is more general and it has both the tangential and normal components. We prove well-posedness and discuss on the regularity results for the control problem. The first order necessary optimality condition results in a simplified Signorini type problem for control variable. We develop a finite element discretization by using $\mathbf{P}_1$ elements(in the fine mesh) for the velocity and control variable and $P_0$ elements (in the coarse mesh) for the pressure variable. The standard energy error analysis gives $\frac{1}{2}+\frac{\epsilon}{2}$ order of convergence when the control in $\mathbf{H}^{\frac{3}{2}+\epsilon}(\Omega)$ space. Here we have improved it to $\frac{1}{2}+\epsilon,$ which is optimal. Also, when the control lies in less regular space we derived optimal order of convergence upto the regularity. We derive a new \textit{a posteriori} error estimator for the control error. This estimator generalizes the standard residual type estimator of the unconstrained Dirichlet boundary control problems, by additional terms at the contact boundary addressing the non-linearity. We prove reliability and efficiency of the estimator.

In this chapter \ref{chapt:Parabolic_Dirichlet_Control}, we propose and analyze an energy space based approach to formulate the Dirichlet boundary optimal control problem governed by linear parabolic equation. We develop \textit{a priori} error analysis for Galerkin finite element discretization of the optimal control problem. We prove well-posedness and discuss some regularity results for the control problem. The first order necessary optimality condition results in a simplified Signorini type problem in three dimensional domain. The space discretization of the state variable is done using usual conforming finite elements, whereas the time discretization is based on discontinuous Galerkin methods. Here, we derive an optimal order of convergence of error in control, state and adjoint state.

Numerical experiments illustrate the theoretical results at the end of every chapter. We conclude the thesis stating the possible extensions which can be made of the results presented in the thesis with some more problems of future interest in this direction.