A Study of Convergence, Quasi-Optimality, and Error Analysis of Adaptive Finite Element Methods for a class of PDEs

Abstract

The central objective of this thesis is to carry out a rigorous analysis of finite element methods for second-order linear elliptic PDEs and for a variety of optimal control problems governed by second-order linear elliptic PDEs. This includes both a priori and a posteriori error estimates, together with a detailed investigation of the convergence and quasi-optimality properties of adaptive finite element methods (AFEM). A key aspect is the treatment of elliptic PDEs in both divergence and non-divergence forms, thereby covering a wide range of practically and theoretically relevant models. First, we focus on establishing adaptive convergence and quasi-optimality results for conforming AFEM applied to second-order elliptic PDEs and Dirichlet boundary control problems where the governing PDEs are in divergence form. Then, we focus on the development and analysis of the C^0 interior penalty (IP) method for optimal control problems governed by PDEs in non-divergence form.

The introductory chapter outlines the essential background on finite element methods and optimal control theory, covering key mathematical preliminaries, a posteriori error analysis, and adaptive algorithms, which serve as a foundation for the subsequent analysis. It also presents a discussion on a posteriori error analysis and adaptive algorithms, highlighting key results on convergence and quasi-optimality. The core of the thesis begins with the analysis of a general non-self adjoint second-order elliptic PDE with a convection term in L∞(Ω). Convergence and quasi-optimality of a conforming AFEM are established under the minimal regularity of the dual problem, namely that the solution to the dual problem belongs to H^1(Ω). The main difficulty—deriving a Cea’s lemma and quasi-orthogonality—is overcome by suitably bounding the L^2-norm of the error in terms of the H^1-norm, using a twisted Schatz argument and the minimal regularity of the solution of the dual problem. Next, the thesis focuses on Dirichlet boundary control problems governed by elliptic PDEs in divergence form. We begin with an energy-space-based approach to the problem governed by the Poisson equation. Using techniques analogous to the Aubin–Nitsche duality argument, we establish the key result of quasi-orthogonality under mild additional regularity assumptions on certain PDEs and a smallness condition on the initial mesh size. Building on this, we prove both the adaptive convergence and quasi-optimality of the conforming AFEM for the corresponding problem. We then extend this framework to a more general setting, where the constraint PDE is a second-order general linear elliptic PDE. By employing the result “uniform inf-sup stability implies quasi-orthogonality,” the main difficulty of establishing quasi-orthogonality is resolved without any additional regularity assumptions or mesh-size restrictions. Consequently, convergence and quasi-optimality of the conforming AFEM are achieved in full generality, thereby extending the earlier results.

Next, the thesis focuses on optimal control problems where the constraint PDE is a second-order elliptic PDE in non-divergence form. We first discuss a distributed control problem and subsequently a Dirichlet boundary control problem, along with their respective formulations. Interestingly, at the continuous level, the optimal costate for the distributed control problem satisfies a fourth-order PDE only in the distributional sense. Similarly, in the case of the Dirichlet boundary control problem, both the optimal costate and the optimal control satisfy fourth-order PDEs in the distributional sense. To approximate the solutions, we develop a quadratic C^0 interior penalty method for each setting. Under minimal regularity assumptions on the optimal state, costate, and control, we establish both a priori and a posteriori error estimates. While the constants in a priori and reliability estimates for the distributed control problem depend on the domain geometry, the corresponding estimates for the Dirichlet control problem are geometry independent, achieved through a suitable choice of the discrete bilinear form and refined estimates of the enriching operator. The numerical experiments included throughout the thesis illustrate and validate the theoretical results. Finally, the thesis concludes with possible extensions and directions for future research.