Research Projects

This project focuses on the Hybrid High-Order (HHO) finite element approximation for distributed optimal control problems governed by quasilinear partial differential equations (PDEs). The HHO approximation is designed and analyzed for stability and convergence using a priori error analysis. The problem is defined on a polytopal domain and considers polyhedral meshes to handle the complex geometry. The design involves a local reconstruction operator that can be solved easily by a simple matrix inversion. The global assembly of this finite element technique is based on static condensation, making the solution procedure robust and efficient. The HHO method is applicable for both low-order and high-order polynomial approximations. The project also establishes error estimations for the discrete state and adjoint variables in the energy norm without control discretization. It then considers a fully discrete optimality system and derives $L^2$-estimation for the discrete control variable. The wellposedness of the discrete linearized problem is used to propose a nonlinear map with a fixed point solution. The existence and local uniqueness of the discrete solution are studied using a fixed point argument. Numerical tests in MATLAB are performed for validation of the theoretical results.