Research Projects

The proposed research develops and analyzes adaptive non-conforming finite element methods for solving optimal control problems governed by elliptic partial differential equations, including both distributed and boundary control settings. The core idea is to integrate the flexibility of non-conforming finite element spaces with the efficiency of adaptive mesh refinement to tackle the challenges posed by low regularity, singularities, and complex control structures in the optimal control problems. The methodology begins by selecting suitable non-conforming finite element spaces, such as the Morley element, which are computationally efficient and well-suited for fourth-order problems. These elements allow for reduced continuity requirements, making them appealing for higher-order equations where conforming methods are impractical. However, this comes with analytical challenges, including the lack of Galerkin orthogonality and the absence of nestedness. These limitations require a careful convergence analysis using enriching and interpolation operator techniques. A central aim of the methodology is to derive reliable and efficient a posteriori error estimators. These estimators identify the local error in the domain and drive the mesh refinement process. The adaptive algorithm follows a standard loop: solving the discrete optimality system, estimating the a~posteriori estimator, marking triangles with high error, and refining the mesh accordingly. To ensure convergence and quasi-optimality of the adaptive FEM, the project employs the axiomatic framework of adaptive finite element methods. This framework is built upon four fundamental axioms: stability, reduction, discrete reliability, and quasi-orthogonality. Together, these properties establish convergence of the adaptive error estimator and guarantee an optimal convergence rate with respect to the number of degrees of freedom. In addition to the classical distributed optimal control problem with higher-order PDE, the research addresses boundary control problems with Laplace and biharmonic PDE, which introduce analytical difficulties due to trace regularity. The methodology incorporates problem-specific strategies to handle these challenges, particularly for Dirichlet boundary control, where non-conforming methods remain largely unexplored. Further, the project extends to mixed finite element formulations, particularly useful when the control functional depends on derivatives of the state variable. This leads to improved accuracy in gradient or flux approximation. The research also considers singularly perturbed fourth-order control problems, where standard methods may break down. In these cases, modified non-conforming methods and alternative formulations are explored, which will be explored for non-linear state equation OCP. Throughout the study, both theoretical proofs and numerical experiments will be employed to validate the convergence and efficiency of the proposed adaptive algorithms. This comprehensive approach ensures that the methods are robust, efficient, and applicable to a wide class of optimal control problems.